Remember that just like kids puppies can get into everything so be aware of things in your house, or even outside your house that could be harmful or toxic to your puppy.
Bulldogs have a lot of wrinkles. Make sure your daily routine includes cleaning the wrinkles and drying them. Doing this will help to prevent rashes, infection, and other skin problems. Baby wipes make cleaning your puppies wrinkles easy. Be sure to use baby wipes that don't have perfume in them. Try to get wipes that are Hypo-Allergenic or perfume free and don't forget to dry them when your done wiping them. Giving your puppy a bath to keep him clean is a good idea as well just remember not to over bath him as you can take the natural oils out of his coat and this can cause problems as well.
You should never leave them unattended in a collar as it is very easy for them to get it caught on something and choke or even hang themselves. Collars and slip leashes if not used properly can also have a hand in causing Cherry Eye. With a harness you can have control with out pulling on your puppies neck. As your puppy becomes a full grown dog he will become stronger and a harness will help you to have better control of your dog without causing them harm.
In the heat of the summer just like humans, bulldogs can suffer from heat stroke. Be sure to provide plenty of fresh cold water, and keep their outside activities to a minimum. Ice is a great summer treat, it will help in keeping your puppy cool on hot days. Not all bulldogs are swimmers. Just like children they should never be left unattended near water. If your going to have your bulldog around water you might want to consider a doggy life jacket but even with the life jacket they should never be left alone.
Like humans bulldogs can have allergies to bee stings, bug bites, different types of food, and even perfumes. if they have reaction give them benadryl. You can apply ice to the area where the dog was stung. If you find that your puppy is having an allergic reaction and is breaking out in hives or has trouble breathing get him to a vet right away.