Train yourself. Ever had any private instruction or coaching? If you’re interested in paddling safely or efficiently, instruction is recommended. Find certified instruction in standup paddleboarding at americancanoe.org. For those that insist on self-learning, here are good videos on sweep strokes, draw strokes, and the cross nose draw.
Train your dog. Start with your board in the yard without a fin. Teach your dog to get on the board, to sit, and to leave the board on command, and then repeat with the board floating in shallow water. It’s best for the dog to have a ‘spot’ on the board to be rather than having him roam around. As Baxter got more comfortable, she roamed the board more and sat less if ever, and as I got more comfortable that became more tolerable. Note that Baxter was canoe-trained this way at two when she got me, and camped and paddled in canoes for seven years before paddleboarding.
Get the right board. Estimated paddler weights for boards are just that: estimates. Try before you buy, and if you can’t try it first buy elsewhere. Boards for dogs need to allow for canine weight and movement. They need extra volume and need to be padded wherever your dog is permitted. Personally I have no preference between displacement and planing hull boards. It’s hard to find an appropriate displacement hull board - I like the Surftech Flowmaster 12’6” but it's no longer sold. The C4 Waterman Pono is an option but personally I have not paddled it. Given that you are already handicapped with some dead weight, a faster design is ideal. Inflatable boards are slower although they provide a suitable deck surface from nose to tail. We did paddle a C4 iTrekker and believe that most would find that less stable than desired with a canine partner. We paddled a 12’1” Laird by Surftech at first but declined to buy it because the deck padding did not go to the nose. That's also no longer sold but this Big EZ, a larger planing hull by Pau Hana seems like the best available option for larger dogs.
Use leashes appropriately. I’d use a coiled leash, attached to a quick release belt at my waist, or knee for smaller dogs. I sometimes worried about Baxter taking off and in those cases I attached a coiled paddle leash to her lifejacket with a skijoring quick release, and attached that to the nose of the board. Because the leashes were attached to opposite ends and were coiled, they didn’t intertwine. Before I switched to a coiled paddle leash I sometimes left her on a standard 6’ leash if it wouldn’t catch on anything when she swam.
Wear a lifejacket. Both of you. I will only use an inflatable PFD when I am responsible only for myself and then only in water warm enough that no wetsuit is needed. Inherently buoyant vests are best if you are responsible for a dog or might need to assist another person on the water. Baxter used a medium NRS dog vest and went through 3 of them in 13 seasons of canoeing and paddleboarding. It was a little small as it was not so much for flotation but sized for where the straps were placed to lift her by the handle. It also wasn’t a full body vest that might have caused her to overheat.
Practice. What do you do if one or both are in the water? You may have the opportunity to figure this out sooner than you’d like.