First off, you'll love mountain biking and are joining a very good community! :) To start, if you have a Giant Rincon, you have a decent entry level bike. I might suggest you work with that until your skills improve.
For a nice, entry level functional mountain bike that don't have limitations against the standard trail with a few rocks, there are 2 things I'd say are absolutely quinetessential: a cassette type hub (at least a 24 speed is the easy way to determine this), and double wall rims, for your safety and the durability of the bike. Also, the more you spend up front, the better bike you'll get... it is MUCH cheaper getting your components on the bike than upgrading.
As a rule of thumb, 500 dollars will get you into this range. Go to some bike stores and hop on some bikes!
Nice aluminum brands that could serve you well in that range would include Specialized (Hardrock, Rockhopper), Trek (4300, 4500, 6 series), Gary Fisher (now a Trek line), Cannondale (F7, F5, Trail SL 3), and Giant (Rincon, Yukon, Revel 0, 1). 29ers (29 vs 26 inch wheels) have their strenghts and weaknesses. They are smoother rolling and will roll over larger bumps with ease. You can go distances with less effort, but they will cost more. However, back to basics, smaller wheels equal more accelleration up slopes and more techical handling. Some 29ers in your range are the Giant Talon, Specialized Hardrock sport disc 29, Cannondale Trail 4 29er, and several Gary fisher Models (note, Gary Fisher's geometry is awesome for some, a nightmare for others as is has a longer, more aggressive cockpit length which isn't always the best for leisurely riding. Disc brakes are nice for stopping on a dime in wet circumstances.
I'd highly recommend one of these brands over a store brand such as Pacific, Mongoose, Diamondback, (exception, they do make some nice models), which you will compromise on the frame. The frame is the core of the bike, which is the part you'll be keeping as you upgrade. So getting a model of bike in each brand that has that nice frame tier jump will make you a bit more futureproof. Trek's don't give you a nice frame until the 7 series, which is much over budget, Cannondales arguably make the best frame, Trail SL, for that price range. Specialized Rockhoppers get a frame upgrade over the Hardrocks, however, it is less durable. Giant Talon 29er is the Aluxx SL vs the regular aluminum.
Ok, I could go on and on.. I'll stop here, but here's the thing... for the money, its a balance of components, weight, and strength. Find a bike that fits, with a reasonably nice and adjustable fork (your most expensive upgrade when it comes to it, hydraulic lockout is nice, so you can turn it off and save your power when on the road while in motion). They each have dramatic differences. But take your time and visit some bike shops.