Before you go spending money please check this:
Will your Wii read or play GameCube discs?
Borrow a GameCube disc from a pal, or buy a cheap one at a game store or an online auction.
If it works with a GameCube disc then the problem is easy to fix if you like tinkering and have a Triwing screwdriver.
Follow the iFixit guides for taking the Wii apart.
Gently work the mechanical disc loading levers - gently - do not force anything.
Look carefully at the two metal guides which pull the full size disc into the Wii. Both should have white plastic clips about 4 by 3 mm to keep them from scratching the edge of the game disc.
Often one or both of these clips will be missing or broken.
Also the metal guides are sometimes bent outward, almost as if you had forced a disc out and it bent them. Gently bend them back towards the edge of the disc. Put a 5mm piece of heat shrink tubing over each of them if the white plastic clips are broken or missing.
With the cover off, try loading a game disc. If needed, bend the guides so they do NOT touch the disc when it is spinning.
This fixes the problem about 1 time in three in my experience.
I had a similar problem -- it would read Game Cube discs, but full-size Wii discs would fail fairly quickly, usually right after showing the title screen. I tried following 8RUC3's advice, but once I got the box open I couldn't find any broken parts. There was a lot of dust, though. After having some fun watching the exposed mechanism load and unload discs, I liberally spritzed the drive cavities with compressed air and put it back together.
It still makes more noise than it should, but the games play now. All for the price of a tri-wing screwdriver and an hour of my time.