I won’t go into all that is involved with leash training as it can be as complex or simple as you want it or need it to be. Most families want to put a leash, collar or harness or their puppy right away and get going. Why would that work so easily? Think about it for a second. The puppy doesn’t know you at all. You are leading them around in a strange place. They are without their siblings. You are setting your puppy up for failure. Please do not do that.
The best thing to do is get your puppy home. Let them have time to get comfortable with you and their new surroundings. Let them want to naturally follow you. What is the purpose of a leash? To keep puppy with you and to follow you right? Once puppy follows you and comes to you when asked, put the collar or harness on. Get pup to follow you again and come when you ask. At that time, add the leash.
Plain cheerios are a great treat for pups. When he or she comes, reward them with a cheerio and lots of praise. This basic leash training of teaching them to come and yield to the leash takes only about 15-20 minutes unless you are impatient.
If pup takes a few steps toward you, pick up the slack but DO NOT pull. Never pull! Ask puppy to come again, praise reward and take up the slack. If you pull, you will create a tug of war type situation. You will cause your puppy to become nervous and to naturally resist and pull back. That is a “no” answer. With training, always ask for things in a way that you can get “yes” answers.
As your puppy becomes older, I highly recommend the gentle leader collar. This ensures that the puppy will never harm his neck or throat, and will save you from ever having your dog out of control from pulling.