I started with the Sizzix Sidekick. It was affordable and has lots of dies. Sizzix dies are often on sale for 40 or 50% off at ACM, M's and J's, or you can use a coupon.
I have successfully cut Sizzlits and Cuttlebug dies, BossKut and QuicKutz dies (with DIY shims). The newer Sizzlits alpha have fewer dies per set with multiple letters, so they are less expensive than the older sets that contained one die per letter.
I got an original Sizzix a couple of years ago for Christmas (on clearance I think). I have a few dies for it, but now want the converter (hard to find) so I can use one machine and expand my collection with some of the larger dies that don't fit in the Sidekick.
Of course I would prefer to get a Cuttlebug and dump both or upgrade to a Cricut (which just isn't in the budget).