Just like collars, you will probably have to buy multiple harnesses for your puppy. I find that harnesses can help with leash training, as your puppy can not choke themselves on their collar. I also find that a well fitted harness is harder to escape from than a collar. There are many styles of harness available. I prefer the style that slides over the neck and buckles on both sides, but I encourage you to try multiple styles on at a local pet store to find what is right for you.
An inclusive checklist for bringing home a new puppy (straight from the mouth of a dog bather, vet tech student, and dog lover all in one). I linked many informational articles as to why you need these things, and what considerations you should make when choosing your new puppy supplies.