First of all, you must not take your puppy to gardens or roads unless he has complete his vaccination. The reason for this is he can easily pick an infection from there. Your puppy will complete his basic vaccination after 3 months after which the boosters will follow. So you can take him out after he completes his vaccination but avoid gardens, rather walk him to concrete roads.
Secondly, you can start leash training right away. You can train him at home. Leash training in the beginning makes your puppy comfortable with collars and belts in the intial stage. If you use belt and collar for the first time and take him out, your puppy may try to take the leash on his mouth and pull it which makes it very frustration for you.
At the same time, if you introduce leash and belt to the puppy at home, he'll be more comfortable when taken out. You can walk him on leash in your house lurring him to follow you on the leash by offering treats. It's a very simple training.