I just shot a 30/30 at my girlfriend's ranch last week and really enjoyed shooting it, but I don't know much about 30/30 rifles regarding their affordability and/or reliability (Affordable to me is no more then $450, maybe $500).
No problem getting a .30-30 in that price range. New Marlin 336Ws (like the 336Cs but with stained birch stocks instead of walnut, less $s) are less than that. Or check the pawn shops. Lots of good deals are out there.
Reliability of Marlin 336s (the only .30-30s I've ever used) are pretty darned good. There are a few known issues with them like the infamous "Marlin Jam", but I've never encountered them. YMMV, of course.
Anyone have an opinion on what kind of rifles I should look at? I'm looking for one that is good for game hunting (deer, elk, axis, blackbuck, etc.) and for target shooting.
I consider .30-30 to be a bit light for elk, but it can be done if you respect the limitations of the rifle. Shot placement is critical, and you'll need to be picky about what bullet you use. Handloading helps. Common wisdom has it that a .30-30 is only good for 150 yards. I'd say that is a bit conservative, if you sight in the rifle at about 3" high at 100 yards it will be about 6" low at 200 yards using factory 150 gr ammo, and that is a usable hunting point blank. Learn your trajectory, and maybe try out the Hornady LeverEvolution ammo which will give you an extra 50 or so effective yards. If you are looking at longer ranges (regularly over 100 yards) I'd suggest getting a low profile, modest magnification scope. My 336A wears a Leupold VX-I 2-7x33 on low profile rings, it seems to be a good match-up for the rifle. No point in magnification over that in that caliber, you won't be shooting that far anyway, and the wide field of view on the low end is a good thing in thick where this sort of rifle shines.
If you learn your rifle's trajectory, get good at range estimation, scope it and maybe use the new Hornady pointy bullets (factory or handload, your choice) you can actually get huntable performance out to 300 yards from a .30-30.
If you decide to go with something with more oomph, then you'll need to troll the pawn shops for something a bit more "mainstream" (not that there's anything weird about .30-30 lever actions -- they are as common as fleas on a dog, and with good reason). A bolt action rifle in .308 Win or .30-06 will do what you want, but you'll need a scope and unless you get a really good deal, well, I wouldn't bet on it being within your budget. Maybe. You can look.