So now that you have a system and few games you'll of course want to play them. Your cartridges will probably have to be cleaned. This is super easy and quick. You'll need a common q-tip and some Isopropyl Alcohol (common rubbing alcohol 91% or better). Dip the q-tip in the rubbing alcohol and rub the moist cotton end on the contacts of the cartridge (Contacts are the shinny metal connectors that look like teeth) Do this a couple times to remove any dust, dirt or old Mountain Dew that may have built up over the game's life. Make sure you change up the q-tip once it gets all cruddy. Also Isopropyl alcohol can leave a residue so be sure not to use too much...you would have to use a lot but I've seen it happen.
I've written a blog series on hooking up old systems so check that out if you get confused between the old systems and your new TVs.
You'll want to find somewhere to store your games. Since you are going to be collecting a lot of these pick somewhere with a bit of space to grow into. Also things with shelves are a plus so that you can see your games without having to dig into them too much. I currently use a storage ottoman from Target, it keeps my games out of the way and seals so they stay pretty much dust free in there. I've drawn up plans to build specialized storage shelves for them in the future, but I don't have the space now. Also once your collection grows a bit you will want to keep it organized so you can find the game you want when you want it.

• Q-tips! Q-tips! Q-tips!
• Some old CD systems will not play nice with CD Buffers (the crank things that remove scratches) Beware!
• Keep you carts stored away from moisture and dust...also don't spill Mountain Dew on them - it doesn't make them awesome like you hope it will :(
• Don't forget you'll also have to store controllers, I recommend "Over/Under Cable Wrapping" this is important so you don't break the wires inside the cable. (Here is a video tutorial)
• Q-Tips!