Congrats on the puppy and on being a first-time dog owner. You're in for a lot of fun! It's certainly never too early to start training. Just keep in mind that he is a "toddler", so you have to have lots of patience while his young brain figures out what you want. You always want to keep training sessions short and fun, fun, fun!
The benefits of training far outweigh the risk of not having his full immunizations. All the puppies in the class will have to be current on their puppy shots. Most puppy classes are held indoors where the floor gets disinfected...a lot.
You may want to carry him from the car to the classroom and back so he doesn't step in (or eat) anything outside the training center that might make him sick.
Keep up the efforts on the nail trimming. In addition to touching his feet daily with your hands, you may want to touch his nails daily with the trimmer (even on days you're not trimming) and treat for him being relaxed. He'll get used to the trimmer pretty quickly if he has a lot of exposure to it.