I just bought a Veritas Rip Carcass saw. It rips wonderfully, and crosscuts are pretty clean as long as I back the wood up to minimize tearout. I've read a bunch of authors state that at around 12 or more teeth per inch (TPI) you really only need rip-cut, and that seems to be supported by my experience so far with this saw. I'd happily use if for cutting every part of a tenon, and I expect I will. I wish I liked the look better (I'd prefer a metal back), but it works quite well.
You mention you want one that will also do dovetails: the carcass saw would be fine for dovetails in large stock, but I wouldn't probably use it for dovetails in, say, a small box. That calls for a smaller saw. Here's what I find I want frequently:
1) A large rip saw. Mine is a 30" Disston, filed to around 7TPI. It cuts anything down to 3/4" wonderfully, though there's quite a lot of tearout on the back. That happens with big rip saws. I think I paid $10-15 for this; your luck may vary.
2) A large crosscut saw. Mine is 11TPI, and I'm pretty sure it's made by Disston, though I can't recall for sure. I'd prefer a lower tooth count (maybe 8 or 9), but it works, it's just a little slower than I want. Again, I use it on anything down to 3/4". This was inherited, but I have a similar one that I haven't cleaned up that I think I paid $10 for.
3) A saw for ripping and cross-cutting thin lumber. I use a Shark brand ryoba for this. It will happily cut through 1/4" stock, and leaves a fantastic edge. I also use it for ripping thicker stock when the board is really short and I need to put it in my vise to support it. I think I paid around $15 for it.
4) A backsaw for handling tenons and small crosscuts with a bench hook. This is the Veritas carcass saw, and I'm really, really pleased I gave in and bought it. They run about $80, and are money well spent.
5) A small saw for really small parts and dovetails. I have an unbelievably cheap and low quality gent's saw for this. I bought it at WoodCraft for about $12, and it pretty much works now that I've refiled the teeth and reduced their set. If you're almost broke and desperately need a dovetail saw, go for it. Otherwise, find a better saw to buy. Probably one of the Veritas dovetail saws would be a much better purchase.
6) A coping saw. I have no idea what I paid for this, but I see reasonable coping saws at flea markets for $5 all the time, and new for $15-$40.
That pretty well covers any basic cutting task I'm likely to need to handle. I have a bunch of other saws, some for sentimental reasons, some because they were bundled with something else I wanted to buy, but most of them aren't even sharp, and only one of them is free of rust. The rest need some serious cleanup. Overall, I have less than $150 in saws, and I don't need any more. I'd like a nice turning saw, and maybe a frame saw for resawing, but those are really kind of specialty items, rather than general use.
