It really depends on what you are wanting to cut. The best thing is to have all of the saws.
Different saws are best for different purposes. A table saw is best for ripping lumber, e.g. making a board that is 5-1/2" wide only 5" wide. A miter saw is good for making precise square and angled cuts, e.g. cutting a 72" board down to 67". A circular saw can perform a lot of the functions of both a table saw and a miter saw, but it's a handheld saw so it's difficult to get the same precision. If you're going to do trim carpentry (e.g. crown moulding), a miter saw is almost essential. That said, there are various blade sizes, as well as bevels and slides. A single bevel, non-sliding miter saw isn't going to cut particularly wide/thick boards, but a saw that does all of the things is going to be heavy, expensive, and may be more than you need. I can do an awful lot with a circular saw, but it takes a bit of skill to use. Think of a piano versus a violin. A piano is big and bulky but, assuming it's in tune, it's going to give you the right note when you hit the key even if you haven't a clue of how to be a really good pianist. A violin is light and small and can make amazing music, but if you don't know what you're doing it's likely to sound awful.
Long story short, from what you are describing, I'm guessing that a miter saw might be a good tool for you, but a double bevel sliding compound miter saw might be overkill. If you aren't in a huge hurry, I'd try to figure out what you want then look for deals around Black Friday. You'll likely need a halfway decent saw with a halfway decent blade (not top of the line necessarily, but most saws will likely come with a crappy blade so figure the cost of a blade into the total cost). As far as nails versus screws, screws are more expensive (but not prohibitively so) but tend to hold better, so long as they aren't visible. Nails are used in a lot of applications where they'll be visible because it's a lot easier to camouflage a nail hole.
BTW, Ana White has a good website for (free) woodworking plans. A lot of them are more basic, accessible plans than you're going to get from Fine Woodworking magazine or wherever, that's going to expect you to have a biscuit joiner and a pneumatic crown stapler. It might be worth looking at some of the plans on Ana White (or just Google "headboard diy" or "coffee table plans") and see what sort of tools people are using to complete their projects.
I'm happy to give more advice as needed.