Just Bought a New Ruger SR 22: What Accessories Do I Need Now?

I purchased the same not too long ago and I have enjoyed it so far.

Have fun and be safe!

ETA:
I use the Otis kit for cleaning, but I have a multiple weapons. Most cleaning kits are fine-it is not required that you spend big bucks on one, just make sure you have a good CLP.

I use inexpensive safety glasses and 3M foam ear plugs at the range.

Cheap FMJ ammo for the range
Some personal defense ammo
Holster

Shutterspeed,
Congrats on your new gun! The main "accessory" you need is LOTS of ammo to familiarize yourself w/ the gun as well as to have a good time. A few other thoughts:
Bulk ammo is fine and is pretty much what I run through my 22's. Also, bulk ammo is frequently very accurate. That's one thing about 22's: there can be a lot of variation from gun to gun as to what ammo a gun likes. If you're just looking for a cheap time, buy the least expensive bulk ammo available. If you're looking for accuracy, buy some 50 round boxes of different brands/types of ammo and test the accuracy before buying a large quantity.

It's good that you're planning on hearing and vision protection. Like lots of shooters, I use the highest rated expanding foam plugs I can find (the max NRR I can usually find is 33). But, I also recommend using external muffs in addition to the plugs. Plugs block conduction sounds through the ear canal only, but not conduction from the bones surrounding the ears; muffs help w/ this. You can usually find an electronic pair from Caldwell for &25. In addition, using the electronic muffs to the plugs will allow you to hear any range commands that you may not hear w/ plugs alone.

Last, you'll want additional mags for your new gun.

CLP and Eezox are my preferred brands of "CLP" products. CLP will probably be easy to find at your local store(s), but probably not Eezox.