What do you need?
You can make rings out of all metal that can be bend by hand and all that can be hammered into shape.
Wire comes in many sizes. Unless you have a hobby involving metal wire, you will likely not stock it. But a good metal shop will have a few kinds and can often order more for you.
No tools to bend:
The 'easy' way is to have wire which is very soft, the disadvantage is that it will not hold shape will when you have your ring.
Hammering and heat/cold treathment will improve that but it is an extra lot of work.
For a cheap and easy ring, get metal that can not be bend easily, get a wooden dowel (or a round metal bar) and fix that into a clamp on a workbench.
Also clamp one end of the wire, it can be in the same clamp if you can get that working, but if needed you can do it with a second clamp.
Now take the long end of the wire and pull it round the bar, till you have a round turn and a bit, that is till the wire lays next to itself for at least a few mm and it does not spring open.
That will make you a ring which will have an opening, which can be closed by soldering or left open, specially if the ends are made showy.
Now you can saw through the wire, close the ring into a circle and solder that shut. (Check with the supplier of the wire what you need to solder, as different metals need different kind of solder and chemicals to make them work.)
If you are an experienced metal worker you can keep the ends longer than just touching and shape them with whatever you want. (Or you can introduce extra materials like beads or stones, or lumps of metal molten to the ends.)
No thick wire?
With thinner wire you can do a series of full turns, solder them all together and have a ring that is wider than it is thick.
A more involved way to make a ring out of thin wire, not for beginner metal workers nor for beginner knot tyers if without an experienced coach, is a turks head knot. Here you find instructions for string. Search results for rings, mmost in silver and gold.
Flat metal can be cut to size, hammered around the dowel and then soldered.
Heat treathment (Warning, dangerous hot metal and fire danger):
Place your metal on stone that can take the heat and apply flame till the metal is red hot. Let if cool down naturally/slowly for a soft metal that will take further work.
If you have a fire which is hot enough, like when you burn coal to heat your house, you can use that.
Cool it down as fast as possible by let it drop in cold water, and it will get harder and less likely to bend.
Hammering will make metal harder but also more brittle and more likely to break if you do it too much.
If you can afford it, silver will give you very nice results, also because the result will be real jewelry. I fear that gold will be out of reach for a while longer, it is rather expensive.
I had nice results with coper to play with but it will colour your skin and you should be aware that it has health risks.
When you buy metal and solder and so on, discuss it with the staff in the shop (if they are knowledgable) or research on internet whether the metal can be used for day to day wear.