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I would recommend renting, or borrowing a tent first before purchasing. At that point yu will have a much better notion of what will work for you and your dog. You are probably looking at a three season tent an you don't want it to weigh more than three pounds if you will use it for any amount of backpacking.
Some prefer a two person tent, even if solo ,for the extra room, but that will mean extra weight or a more expensive purchase.
A tent really shines when you encounter rainy,or snowy, weather. A bath tub floor, in which the floor material extend sup the sides of the tent a few inches is a desirable feature.
Beware of really cheap tents. They are a waste of money.