Papayas are an excellent fruit to grow, since they fruit pretty quickly and year round.
Papayas can be grown from seeds, and you can actually use the seeds from shop-bought papayas! This is certainly the cheapest method, and possibly also the most enjoyable.
You should be aware though, that even if papayas are easy to grow, the plant also tends to die easily, so you will need to care for it, especially during its early stages.
Papayas are native of tropical climates, so you will obviously have an advantage if you live in a tropical of sub-tropical zone. In general papayas need lots of sunlight, lots of water, fertile soil, and absolutely no frost. If you live in a zone where the temperature drops often below 30°F, your plants will likely die.
To get the seeds, just cut a store bought papaya in half and take out the seeds. They are usually very abundant and one papaya will likely provide you with enough seeds to fill your garden and even your neighbors'.
The seeds should be planted in your garden (pots are no good, especially when you consider the trees grow up to 20 feet in height), in a spot that is sunlit for at least 6 hours a day.
Beware the transplanting processes as well: their roots are very fragile and do not respond well to damage, so it is best to plant the seed where you want the plant to grow.
Papayas produce lots of fruit, but to do so they also need lots of nutrients from their soil: use a lot of good compost and soil, possibly a meter across, and plant about 20-30 seeds in your patch. Cover the seeds with half an inch of compost, then mulch them well, and wait a couple of weeks.
You will notice that not all the seeds produce healthy plants: some of them will inevitably be stronger and larger, and those are the ones you want to keep!
You should keep about 6-8 plants, since papaya plants can be either male, female or bisexual: the male gender does not produce fruit, so you want to be sure you're keeping at least one or two females and bisexuals.
Your plants will start flowering when they have grown to about a meter, with male plants flowering before the others. You can recognize them because male flowers have several blooms and thin stalks, while female plants have a single large bloom, and short stalks.

A good ratio for pollination is one male plant for every 10-15 female plants, so cull most of the males.
Watering papaya is a bit of a dilemma: they need lots of water, and it tends to evaporate a lot, especially during the hot season. At the same time though, their roots are fragile and tend to rot easily, so you're faced with the difficult task of balancing the amount of water. Most growers tend to overwater them and cause the roots to rot, so try not to overdo it!

Due to their high yield, papaya trees need lots of fertilizing, especially nitrogen. Manure is a good choice since it provides most of the nutrients they need, and you should also use lots of compost and mulching, especially as the plant grows bigger.
As for sunlight, just follow a simple rule: the more, the better! Even if leaves have little brown areas due to sunburn, don't worry, since papayas just love sunlight! More light means more fruit, and better-tasting at that, so choose a very exposed patch.
As I've already mentioned, papayas fruit all year round, even though they will likely stop flowering and fruiting if the temperature drops too low. This is only temporary though, and they will resume their activity when the temperature warms up.
In general, younger trees are better for many reasons: the most practical is that it's difficult for you to pick papayas 3 meters in the air, but trees also tend to become weaker and produce less fruits as they age, so the ideal way to handle the problem is to keep planting new trees every now and then.
Most papaya plants die in their third or fourth year, since they become susceptible to several diseases, but if your plant is sturdy enough it can actually live for several decades, although fruit yield will likely be inferior.
Papaya trees may suffer from several problems, including root rot, diseases and pests, and blowing over caused by winds.
You probably guessed that their weak spot are their roots, since the tree tends to become very tall and unbalanced, and strong winds will probably kill it.
If you're lucky, replanting the tree will save it; another alternative if it's old and tall is to just cut it at 2-3 feet high. Sometimes they die, sometimes they grow back and produce more fruit that you can easily pick without ladders.
If you decide to cut the trunk, make sure you're in the dry season, since the trunk will rot if overwatered.
As for diseases, trees tend to get them when they age, and growing new trees is so easy you shouldn't worry about diseases and just let the old plant die. The secret with papaya plants is replanting regularly, that way you get a constant flow of fruits and healthy plants.