I just read your post more carefully. Obviously you have a lawn. Composting can be very simple or very complicated, depending on what you want. Sounds like you want simple for now.
If you want a simple setup you basically need two things:
A container that will loosely hold your compost and some stuff to rot.
First the container. There are lots of commercial options but really, any sort of decently tall box-like structure will do. Many make them out of old pallets. Super simple and cheap if you can find them.
Next is the stuff you put in it. For an easy compost situation, you just need to keep some things in mind.
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Avoid meat. Your compost will not get hot enough for this in all likelyhood.
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Avoid anything with seeds if you plan on using your soil for gardens. Again, it probably wont get hot enough to kill them and they will grow in your garden.
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Lots of brown stuff! Think leaves. More leaves. And even more leaves. This will bulk your compost up.
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Careful with weeds. Its tempting to throw weeds in the compost but again, if you are using it for gardens it might end up just growing a crapload of weeds.
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Add some green matter. Think salad leaves, stems, etc. The only thing to worry about here is to be careful not to throw anything in you think will attract a ton of animals.
And thats pretty much the basics. You can turn it from time to time but you dont HAVE to. If you live somewhere dry you might want to water it from time to time too but if you get rain it should be fine.
Remember, at its simplest youre piling stuff up to rot. Nature does the rest. You can get into the nitty gritty later. For now, just start and experiment.