Plants have an enormous number of ways of propagating.
Very broadly they can be broken into several overlapping categories that are not mutually exclusive: vegetative (vegetative propagation is generally/always? asexual), seed/spore based, asexual, and sexual.
Vegetative reproduction is common, meaning that the plant breaks off a piece of itself or clones itself. Stolons/runners (at aor just below the surface - think strawberries), rhizomes (below the surface, think bamboo or ginger), adventitious root (think ivy), bulbs, corms, & tubers (think chives when the bulb divides, or dutchman’s breeches, or potatoes), tip rooting (think brambles or grapes), broken limbs or fallen trees rooting (think willows and redwoods), dividing (think creosote and curly parsley), formation of small plants ready to root (think spider plants and mother-plants), is a non-exhaustive list of potential vegetative ways of reproducing. These methods are often non-exclusive as well, meaning that a plant may use more than one of these at a time. These are generally asexual methods of reproduction that produce what are essentially clones of the original plant.
Both sexual and asexual reproduction can produce seeds/spores/etc, and when this happens there is generally some genetic recombination that takes place. This can be relatively straightforward (flower, pollination, fruit/seed, seed falls, germination, new plant), or can be wonderfully complex and bizarre, as in ferns. Generally asexual propagation doesn’t lead to seeds, but some species can produce seeds this way, it’s called apomixis, when they do, and self-pollination is not considered to be asexula reproduction, that’s a subset of reproduction.
When it comes to sexual reproduction there are a few more factors to consider. Pollination (their fertilization of the reproductive structure), and plants have a variety of methods for this, but they basically come down to self pollination, wind pollination, or co-opting animals to to the pollinating, and the plants will make all sorts of specialized structures to appeal to (or trick) specific pollinators.
Similarly, when it comes to dispersal of the viable reproductive structures plants employ a wide range of strategies; some just use gravity, others wind, some use water, many hijack animals via a variety of methods (bribing with fruit, having sticky or clingy seeds - called phoresy -, sacrificing some seeds so that others survive, etc, etc).
In short, there isn’t a brief, simple answer to how plants propagate, they use a variety of methods, and a single plant may employ more than one at the same time, or depending on environmental situations.
