When you turn the key to the ON position, you should hear a high-pitched whine from the CDI. Check the connection to the CDI box before assuming that the box is bad. Check that you're getting spark, ideally on all cylinders.
You're looking for a strong blue/white spark that should jump a good 15-20mm from the plug to ground. I've seen your symptoms on a car with a new coil that proved defective. In that case, the spark was more yellow/orange and didn't hurt as much when I put it to my tongue.
When you pull each plug to check for spark, immediately snap a photo, and show us the two shots that look most different.
The engine needs to turn over quickly when you turn the key. If it's not, you can skip checking for spark. I'd start by making sure you have a solid ground between the tranny and the chassis; just use jumper cables or a new ground strap. If that helps your starter speed, it may help your spark issue as well.
Finally, try to limit the changes you make while debugging this problem. You will get plenty of advice from the forum, but the more changes you make, the greater the chance that the problem is complicated but new issues and the harder it will be to identify.