Buzzing on the lower frets either indicates back-bow, or simply not enough relief on the neck. (It should be bowed slightly forward, depending on how low you want your action and just personal taste.) If it's just buzzing at the nut, the nut may be cut too low.
Go to www.ibanezrules.com
and check the tech pages he has there for detailed information on how to properly set up your guitar. It's worth the effort.
I got my first electric guitar, an Ibanez, a little over a year ago and was totally clueless how to set one up. My guitar was used and out of the box I could tell it wasn't right so was already a member here and in my 'New Guitar Day' thread asked some advice, but then remembered reading that Rich (owner Ibanez Rules) has a long, detailed tutorial about setting it up.
Now it's like my 'Setup Bible'. I've tweaked it a bit for my taste, but it was a great starting point and it teaches you what to look for, what to do and what not to do.
This is a fixed bridge guitar, right? Going out of tune may be due to strings not stretched quite enough. Did you put new strings on? If you haven't, I'd suggest doing that first thing. Also; don't leave too many windings on the tuning peg, 2 or 3 at most. And stretch those things out.
Also make sure everything is tight. Tuners, bridge mount, neck bolts, etc.
If you know all this already, please; sorry for the assumption that you don't, but what you're describing is exactly the things I didn't understand about setup at first.
I'm just a dumb-blonde, and I got it all sorted out.