Welcome to YT! I bathe Scooter about every other week. I like the puppy cut, just like you. It is really hot here most of the time so the puppy cut really keeps him cool. His hair grows way faster than mine so he gets a hair cut about once a month at this point. With his hair being short, I use a 2-in-1 shampoo and conditioner from Petsmart. Nothing special but it does the job. I also like to have puppy wet wipes on hand for quick clean-up. For example, if it gets muddy outside or he gets in the bushes, I take out a wet wipe and within seconds he is clean again. Wet wipes are also good for cleaning their rears when they need help.
You may also want to get the wipes for eyes. Scooters eyes water and make tear stains. The little pads are perfect and sensitive enough for them. I use them maybe once a week.