How Many Skills/tricks At A Time For A Puppy?

I sometimes teach 5 or more tricks at a time. But, do remember to complete at least some of them. The purpose of teaching is for them to learn, and if you get too overeager to teach more and more tricks you'll have a dog that only kind of knows a few tricks, and doesn't completely know any.
If it's confusing for you to work on 4 tricks all at once, then don't do it. If you're confused, he's definitely going to be confused. So figure out what's comfortable for you and for him and go with it. If it's a pretty complicated trick, for the sake of both of you, work on it with easier tricks or just by itself. In other words, don't try to train playing chess, weaving, putting toys away, and getting a drink from the fridge all at once. In my opinion it's tough and mentally exhausting for everyone involved. If I'm training a tough trick, I train it with easier tricks so I don't frustrate myself or the dog.

Also remember that just because Peanut knows the trick at home, doesn't mean he knows it at the park or at Petsmart. Everything at home is old; he's used to it. Nothing's distracting. The park, Petsmart, etc; these places are packed full of distractions and listening to you is so boring compared to them!
Come and stay. These are two very, very, very important commands that can keep Peanut out of a lot of trouble--if you make sure he can obey these commands anywhere. When he gets to a new place, you basically have to pretend that Peanut has lost his mind and has temporary amnesia of all of his training. With a high value treat(bits of chicken, hot dogs, or cheese; rolled dog food, anything that really gets his attention), ask him to do a simple trick. Let's say you've asked him to sit. If he does, YAY! Reward. If he doesn't, act like you are having to teach him again, like it's a brand new trick. Don't worry, this go round it won't take very long. He just needs a refresher.

What I'm saying is that you might also spend time advancing the cues he knows best to new, distracting environments. Good places to start are low-traffic parks, parking lots, where he will see people or animals but you can stay fairly far away from them; and hardware or hunting stores that allow pets but may not be as busy as Petsmart/PetCo. If your Petsmart/PetCo/other chosen pet-friendly store is really slow at a certain time of day, call to figure that out so you can take advantage of those times. You don't want to immediately start with high levels of distraction (i.e., starting at Petsmart on a Saturday--probably way too distracting for Peanut).

Hope this helps and good luck with little Peanut!