Ibanez GRGA11. I have fret buzz on almost all frets if I play them hard enough, and open strings are buzzing too. The gap between the 7th fret and the string is ~3mm. Is this something I could fix at home, or should I take my guitar to a tech?
Do you hear the buzzing through the amp or just unplugged? A lot of guitars will buzz some unplugged, but unless you hear it when amplified it isn't a big deal.

This is something you can fix yourself. There are tons of websites that show you how to set up a guitar. Google guitar setup, and take the time to learn to do it. Here is an excellent site http://www.fretnotguitarrepair.com/repair/electric-guitar/buzz.php
Do they buzz when you play "normal"? It's an electric guitar, so you don't have to really whack the strings - just turn up the amp

I'm getting at what Stormin is saying. If there's a slight buzz but you don't hear it through the amp, it might be OK. I think it can be a tradeoff that you want really low action, but will tolerate a slight amount of buzz when you get carried away. Check those web sites and raise the bridge or bridge pieces (depending on the design) if it bothers you.

You might need a truss rod adjustment, but that's a little more advanced than just raising the strings a little when you're just starting out. You want to do some research if you think that's your problem.
Since timbo63 mentioned adjusting your truss rod, let me comment on that. The function of a truss rod is to adjust the relief (curvature) of the neck. While adjusting it will have an effect on string height, that is not its purpose. Most necks are set to have a slight relief.

You can check this by holding down the string at the first and last frets, and seeing if there is any space between the top of the fret and bottom of the string at the midpoint (usually about the 7th or 8th fret). If there is a space about the thickness of a thin guitar pick, the relief is OK.. don't mess with the truss rod.... any adjustments for string height should be done at the nut and/or bridge.

If the string is touching at the midpoint it means you have no relief or possibly even a back-bow. You may want to loosen (counter-clockwise) the truss rod. Give it a quarter turn, leave the guitar for an hour or so, recheck, and repeat as necessary. This adjustment may raise the string action, which should be adjusted at the bridge if needed.

If there is a space greater than a thin guitar pick you may want to reduce the relief by tightening (clockwise) the truss rod. This may lower string action... adjust at the bridge if needed.

When setting up a guitar, neck relief should be checked first and adjusted as needed. Once it is set right, leave it alone. Any further adjustments should be done at the nut and/or bridge.

Here is a picture to show neck relief