The critical steps for any sexually reproducing organism can be summarized in a few sentences. Sexual reproduction involves the combination of two sets of chromosomes via the fusing of two nuclei, each of which contains one set of chromosomes. Genes reside in those chromosomes. A nucleus with one set of chromosomes is called a haploid nucleus, the haploid nuclei are contained in cells of some sort and naturally a cell with a haploid nucleus is called a haploid cell. The first step towards fusion of the nuclei is the fusion of two haploid cells and this step is called plasmogamy. At some time after plasmogamy the two nuclei meet and fuse, a process called karyogamy and this yields a diploid nucleus, namely one with two sets of chromosomes. Later new haploid nuclei are formed, either from the original diploid nucleus just described or from copies of that original diploid nucleus, via the process of meiosis (which you can find described in numerous genetics text books or websites). During meiosis there is mixing and resorting of the genes so that the new haploid nuclei are genetically different to the original haploid nuclei.
That summarizes the essential processes but there is much variation in the ways those processes are carried out by different species and their timing. In humans the haploid nuclei are carried within sperm and egg cells, collectively called gametes. Very soon after plasmogamy (ie, fusion of a sperm cell with an egg cell) the sperm nucleus moves into the egg cell and karyogamy occurs. The now-diploid egg cell gives rise to an embryo and ultimately a new human individual develops. Humans are diploid-dominant organisms, that is to say the great majority of cells in a human body are diploid cells in which each new cell contains a copy of the original diploid nucleus. Eventually haploid nuclei (and gametes) are produced by that new human individual, via meiosis, from copies of the original diploid nucleus. Human sexual reproduction requires two individuals, one to produce sperm the other to produce eggs, and humans are incapable of self-fertilization. However, sexual reproduction doesn't always demand distinct parents and many people are likely to know of plants capable of self-fertilization. Note that a plant capable of self-fertilization is not restricted to self-fertilization. There are many heterothallic fungi (including lichen-forming species) where sexual reproduction must involve two individuals. There are also homothallic fungi (including lichen-forming species) which are similar to the self-fertilizing plants mentioned above, in that sexual reproduction doesn't demand two individuals.
Sexual reproduction in lichens is part of the broader subject of sexual reproduction in ascomycetes and basidiomycetes and much is known about the processes of sexual reproduction in the non-lichenized species. Ascomycetes produce their sexual propagules (called ascospores) within microscopic organs called asci and basidiomycetes produce their sexual propagules (called basidiospores) on microscopic organs called basidia. The ascospores or basidiospores are typically haploid and so have similar roles to those of human gametes - except that spores don't fuse. Mycelia are involved in that process. It is possible to germinate the spores and grow the mycelia of many non-lichenized fungi in the laboratory and study the entire cycle through to the production of the new haploid nuclei and sexual spores. Such studies have shown one major difference between humans and fungi. In the human reproductive cycle we can say that the male function is to donate a haploid nucleus (via the sperm) and the female function is to accept a haploid nucleus (via the egg). Moreover a given human can be either a donor or an acceptor but not both. Donation and acceptance of haploid nuclei also occur in the ascomycetes and basidiomycetes but a mycelium may function as both donor and acceptor. Furthermore, in the case of a heterothallic species one mycelium can accept a nucleus from another provided the two mycelia have different, genetically determined mating types. How many mating types are there? It varies with species and can range from two to thousands.
Unfortunately, very little is known about the corresponding processes in the lichenized fungi. The fungal partners can be grown, to some degree, in the laboratory but not to the extent that would allow a good understanding of all the processes of sexual reproduction. A variety of macroscopic and microscopic features (such as apothecia, perithecia, asci and basidia) are found in both lichenized and non-lichenized fungi. This suggests that the processes that give rise to such features in the lichenized fungi are similar to the processes that give rise to the same features in the non-lichenized fungi - but until there is proof it cannot be assumed that the lichenized fungi mimic the non-lichenized species in all processes, as the following paragraph will show.
In many non-lichenized ascomycetes there are distinct nucleus donor and nucleus acceptor organs, respectively antheridia and ascogonia. Since a mycelium could act as donor or receiver, it might be producing ascogonia in one place and, at the same time, antheridia in another - or the timing might be different. The transfer of nuclei from antheridia to ascogonia is via trichogynes, a trichogyne being a filamentous-like outgrowth from an ascogonium to a nearby antheridium. There may have been multiplication of nuclei in both the antheridium and ascogonium, so that numerous nuclei pass through the trichogyne, or there may be multiplication of both types of nuclei in the ascogonium after the antheridium-ascogonium connection has occurred. Regardless of the process, at some stage after the connection nuclei will pair off in the ascogonium. Each pair will contain one ascogonial nucleus and one antheridial nucleus, but there is no fusion of nuclei (ie, no karyogamy). Hyphae grow out from the ascogonium, each hypha bearing one of the pairs and a pair is copied into each new hyphal cell. Such hyphae, where each cell contains two genetically distinct nuclei, are neither haploid nor diploid but are termed dikaryotic. Here we see another difference between fungi and humans since the dikaryotic state is absent from the human reproductive cycle. The dikaryotic hyphae eventually give rise to asci and it is in the rudimentary asci that karyogamy takes place and diploid cells are formed. However, in any such rudimentary ascus the diploid state is very short-lived since the newly formed diploid cell quickly undergoes meiosis to produce new haploid nuclei. These become the basis for the development of new spores which will mature as the ascus develops. There has been considerable study of lichen anatomy but antheridia have never been observed. In the non-lichenized ascomycetes fertilization via an antheridium is not always necessary. Many species of fungi produce vegetative propagules called conidia (which may be haploid or not) and these may germinate to produce a new mycelium. Various fungi produce tiny conidia, seemingly with a primary role as fertilizing agents (much like a human sperm), rather than as the sources of new mycelia, and such conidia are sometimes referred to as microconidia or spermatia. Note that even a haploid (macro)conidium or a haploid ascospore could act as a fertilizing agent. In many lichens ascogonia have been seen to form near the thallus surface with trichogynes growing from the ascogonia and protruding beyond the surface. Many lichen species produce microconidia and these have been seen on trichogynes, where it is likely that those conidia are acting as fertilizing agents. However research is still needed to confirm these suppositions and to reveal the precise processes. As recently as 2008 it was noted that:
Migration of conidial nuclei through the trichogyne and pairing with ascogonial nuclei likely occurs, but has never been documented in lichen-forming ascomycetes. In many species ascogonia were never found with protruding trichogynes;...others produce large numbers of trichogynes but neither macroconidia nor microconidia, which might serve as gametes.
To complete the account of nuclear donation and acceptance I'll note that in various non-lichenized ascomycetes donation-acceptance of nuclei can occur after hyphal fusion and that the basidiomycetes never produce any distinct donating and receiving organs. Instead receipt or donation of basidiomycete nuclei often occurs after hyphae from two mycelia have come into contact and fused. A haploid conidium or haploid basidiospore could also act as a donor to a haploid mycelium that comes into contact with the conidium or spore, assuming there is mating type compatibility between the mycelium and the conidium or spore.
Dikaryotic hyphae are common in the basidiomycetes and mycelia composed of dikaryotic hyphae are formed when two, compatible haploid mycelia meet. For the non-lichenized basidiomycetes this happens out of sight within some substrate (for example soil, wood, dung) and the resulting dikaryotic mycelium also remains out of sight. It is potentially long-lived and will produce the visible fruiting bodies (such as mushrooms, puffballs, bracket fungi, etc). In such fruiting bodies the basidia are produced and it is in the rudimentary basidia that karyogamy occurs. As in the rudimentary asci the diploid state in each basidium is very short-lived since the newly formed diploid cell quickly undergoes meiosis to produce new haploid nuclei. These become the basis for the development of new spores which will mature as the basidium develops.
You have seen that not only are there significant differences between the sexual reproduction processes in humans and those of ascomycetes and basidiomycetes (for example the existence of the dikaryotic state in the fungi) but there are also significant differences between the processes of ascomycetes and basidiomycetes. Many of the former have a limited dikaryotic stage within the fruiting body whereas in the latter many have a free-living dikaryotic stage. There are still more complexities of sexual reproduction in non-lichenized ascomycetes or basidiomycetes, for example the existence of multinucleate (rather than just mono- or bi-nucleate) cells or dikaryotic spores in various species - to give just two examples. Therefore there is potential to find yet more processes that occur in the non-lichenized species but not in the lichenized ones!