Before choosing a modchip it is important to know what chipset your wii is. Entering your wii's serial number here http://wiitracker.nintendo-scene.com/search.php will give you a pretty good idea what chipset you have. If you cannot do this for some reason then there are a handful of chips that support all chipsets.
If you have an earlier wii, then a wiikey is probably the way to go. But the Wiikey only works on DMS, D2A and D2B chipsets. If you have a D2C, D2C2 or a D2E then you need a different modchip. D2Pro 9 wire, wiikey2 and the newest wasabi version support all current chipsets. Also, if you have a D2C or later you don't have to worry about cut pins. Only some D2B chipsets had cut legs. Cut lets made it more difficult to mod the console.
If you are bad at soldering then something to keep in mind is a wii-clip. Solder the clip to your modchip and then you wont have to solder anything to the console. It makes an install MUCH easier, though does have the drawbacks of coming loose and being generally less reliable.
At the current time THERE ARE NO WII FIRMWARE UPDATES THAT CAN DETECT MODCHIPS. This may change in the future but for the time being it stands. Some newer games can detect some older modchips (giving error 001) but this can be fixed on some modchips by updating the chip's firmware.
You can safely allow the console to do updates from a game so long and ONLY SO LONG as the game you are updating is from the same region as your console. Updating with a firmware from a different region can brick or semi-brick your console and should never be attempted.
A semi-bricked console can be fixed by running a later update than the one that bricked the console from the proper region. Example: You have a NTSC console at firmware 3.1 and update it with the PAL version of 3.2. The console is now semi-bricked. Games still play but the wii settings cannot be accessed. If you run the NTSC 3.3 firmware it will fix the console.
Questions on firmware 3.3:
Wii firmware 3.3 wet briefly blocked the twilight hack (not gonna explain what that is here, if you don't know search for it) but the hack has been fixed now and will work again. And it blocks truncha signed disks. Truncha disks can still be run by loading them through the gecko OS on the homebrew channel. Scrubbed and brickblocked disks still work. If you want more info on truncha and scrubbed disks search for them. As was said earlier, NO FIRMWARE UPDATES CAN DETECT MODCHIPS, AND THIS INCLUDES FIRMWARE 3.3.
Theres also a second update to firmware 3.3. This one blocks wad installers and the homebrew channel installer, so if you have this newest firmware and do not have the homebrew channel already installed then you're out of luck for now and will have to wait for new installers that get around this. Some reports say that if you had the custom IOS firmware installed before this new update then you can still install wads and the homebrew channel, but this update is still very new so reports are mixed.
Online play:
Online play with a modchip and a backup copy of a game works exactly the same as it does on an unmodded console with an original copy of the game. It works just fine and you won't get banned.
Gamecube game backups can be safely run from any region without fear of bricking because none of them have updates. They can also be burned on normal sized DVDs, and more than 1 can be burned to a single disk using GCOS.
Updating the modchip:
Not all modchips can be updated. There is a very nice for updating the wiikey here http://forums.afterdawn.com/thread_view.cfm/597460 but some D2C chips cannot be updated. Others lack the ability to update from a DVD and instead use a USB ribbon cable that connects to the PC. Not going to go into all the details here. Again, if you need the info try searching for it.
Backing Up Games:
First off is downloading or backing up your games. If you choose to download one you will most likely get a collection of rar files. There's a guide on what to do with them to get the iso file here http://forums.afterdawn.com/thread_view.cfm/555919
Backing up your games is a little more complex, but there are 3 ways to go about doing it.
First off remember that most DVD drives cannot read wii disks. In fact, only 4 models of DVD drive can read them, and they can be somewhat hard to track down.
The second option is to use an SD Gecko, its a gamecube memory card with an SD slot. If you have a modded console you can run a homebrew program that will dump an iso of the game in your wii console to the SD card in the Gecko.
Finally there is a program run through the homebrew channel that can dump games from the wii directly to your PC via wifi. I really don't know much about this program beyond it requiring the custom IOS wii firmware, and no one I know has even attempted it yet.
Burning Games.
Once you have your iso image the process of burning the game is simpler. Any DVD burner will work for burning wii games. Simply load up imgburn or nero and burn your iso file. I recommend burning at 4x to reduce errors. The brand of the disk used is also important. Verbatum DVD-R is preferred by most, but +R is fine as well. Don't bother with the off-brand disks as you'll probably end up with a bunch of coasters.