Just got a 360 over Memorial Day weekend.  So far I have Hexic HD (came on the console), TMNT 1989 arcade, Aegis Wing and Street Fighter IV.  Burnout Paradise and Banjo-Kazooie Nuts n Bolts are on their way to the house.  I have three questions:

What have you recently been playing pre xbox, and what sort of genres/specific games do you like?

I've been playing Need for Speed: Most Wanted lately, which is like... a launch title. It's still really good though, racing.

Cheapest way to do  wireless would be something like this

Bionic Commando: Rearmed is $5 on XBL this week only. You should check it out.

wow, lots of replies super quick.  thanks!

Not sure if you're looking to play a game with the kid, but I was surprised by how much fun Spiderman: Friend or Foe was.  Simple sure but if you want a co-op game to play with a younger kid it's good.  Played it with my little brother and he usually struggles with stuff I tried to play with him but we had a good time with that game.

I connect to my wireless router by bridging the ethernet and wireless connections on my laptop and connecting my Xbox 360 to my laptop via an ethernet cable.  Not sure if that's an option for you but that's how I do it.

Old cheap games to pick up IMO: Mass Effect and Crackdown.  Don't think either are too violent and both are a lot of fun.

Thanks Stephen, yes, I'm using internet sharing on my mac and connecting the mac to the internet via my wireless router so I'm doing something similar; I was just looking for a solution that wouldn't require my laptop constantly be used by my xbox (again, toddler alert).

Banjo-Kazooie and Banjo-Tooie! :)

In my eyes, some of the best in the "genres" in my eyes:

Lost Odyssey- Phenomenal RPG, excellent voices and great story, however, can lose steam a bit closer to the end before it picks back up for the ending.
Banjo 360-Action/Vehicle/Lego/Platformer?- Great game, with such a belcurve of a learning curve. Can be frustrating, but when you do well, you feel like a genius
Eternal Sonata is also quite nice, gorgeous visuals and Chopin music.

On XBLA, all I can think of is SSF2THDR and Bionic Commando:Rearmed. Both quite hard however.

Some suggestions for great, affordable games:

Retail: (should be able to easily find new copies under $25)
Crackdown
Viva Pinata:TIP

Arcade: (400 pts)
Uno**
Feeding Frenzy
**Uno is a hoot.  Tons of people still playing, easy way to meet folks on live.  Just don't enable the camera if you have one...

Arcade: (>400pts)
Puzzle Quest
Poker Smash
Pacman C.E.
Peggle
Pinball FX
Space Invaders Extreme

You should really try out Rez HD. It's an arcade game, so there's a free trial you can play. It's a on-rails shooter, but it's really cool artistically. Music is real good too. No violence or language whatsoever. E. Geometry Wars: Retro Evolved 2 and Pac-Man Championship Edition are also good choices. The prices of all these games should all be around $10 each.

As for disc games. Mass Effect. Yeah, it's M-rated, but it's a light M. There's no real heavy cursing or violence or anything like that. It's a sci-fi shooter RPG. So if that sound appealing to you, I highly recommend it. I think there's a package out there where you can buy it for $20-$30 and get the DLC pack that it came out with for free. If you're into cool character interaction and stuff in games, Mass Effect is awesome. My personal favorite game on 360.

If you're into simulation racers. Forza Motorsport 2 is really good. It's more fun than something like Gran Turismo because you can customize just about everything on your car and give it custom paint jobs and everything. Pretty cool game if you're into that. You can get that for $20.

And of course there's Halo. I don't know if you want something like that though. It's M-rated, but isn't that violent or vulgar. Just an epic sci-fi FPS.

If you have a laptop then you'll be able to connect to xbox live using that.  You'll have to go in and change some settings on you laptop to get it work though.  I found out how to do that from a youtube video.

The original Puzzle Quest is a heck of a good game for the money, and if you like Hexic, it's kind of in the same vein.  The Wallace and Gromit games are decent, too.

For regular disc games, with a toddler in the house, you can't go wrong with either Viva Pinata.  Others you might want to check out include: Rock Band/Guitar Hero, Fable 2, Kameo, Forza 2, and most of the Need for Speed games.  Those are all pretty kid friendly.

Looks like Viva Pinata, crackdown, M:UA, geometry wars and bc:rearmed are my winners for now.  I will probably look into Mass Effect when I want something a little more involved, storywise.  I had no idea that Wallace and Gromit games were already out, will look that up too, as I'm a huge fan.