At 12-16 weeks of age, your puppy will be growing rapidly, both physically and mentally. Puppy's senses and motor skills are becoming more advanced, which means you'll notice his curiosity peaking, a more acute awareness of his surroundings and less awkward movements as he continues to discover his new environment and practices walking and running. A major development is something called "flight ingraining," which means that instead of following you around everywhere, puppy is now starting to test limits by exploring boundaries. Puppy is excited by his newfound independence and you may notice him always headed in the opposite direction of you. Don't let it become a habit; consider this the time to reel him in with both leash and obedience training!
Congratulations on the progress you've already made in potty training. It should be getting easier as your puppy gains better bladder control, though you should still expect random (and sometimes, frequent) accidents to occur. Remember, patience and consistency is key to house training success. You may also want to enroll your puppy in a group training class to start socialization training if you haven't already. Between 0-16 weeks is by far and away the most crucial learning period in your pup's life. Consider him a sponge for soaking up your teachings. While there will be other opportunities to refine and change behavior, this is the time period when training is the most successful.
Your puppy should have already received his first vaccinations, but should be getting boosters at 12 and 16 weeks. Make sure you're up to date on your immunization schedule by checking with your vet. For example, Parvo is typically given at 16 weeks. Vaccines are required for most pet service providers such as groomers and daycares as well as dog parks, so it's imperative you're current on those shots before putting your dog or another's dog in a potentially dangerous health situation.
Make sure that your home is not only puppy-proofed, but that you keep small, chokable/swallowable items away from your pup.
Your puppy should be eating high quality, solid food now that he's been weaned from his mama. Check with your veterinarian on the right feeding plan (on both amount per serving and frequency), but dependent on the breed, your puppy will likely require a few smaller meals throughout the day than adult dogs. As he grows, his "puppy teeth" will begin to fall out and be replaced by "permanent" or adult teeth. Just like a human baby, puppy will want to teethe, so make sure to give him plenty of chew toys to soothe his sore gums (and to prevent unwanted chewing on valuables!).
At this age, you'll want to introduce your puppy to all kinds of new experiences to make sure he starts to get comfortable around not only other dogs but also various kinds of people - men, women, kids, senior citizens and strangers. At around 16 weeks is often when your puppy enters a fear stage. You'll want to create positive associations with new things that may be frightening to your puppy such as loud noises or other animals to foster a healthy, safe transition and assimilation to his new world.