Considerations;
So, can you afford a dog? Pet insurance (I pay £15 a month for my healthy young dog who has never had a claim- plus excess of about £70 if I did claim). Some dogs may need professional grooming. Training classes. Plus adoption or purchase fee. Toys, food, etc.
Do you own your own home? If no- lots of tricky considerations. If yes, not so hard. Secure the garden and consider that a dog (esp. a puppy) may chew things, get hair everywhere, wee on the floor once in a while, etc.
How much time do you have? You don't have to be interacting with your dog all day long but you need to be able to give the dog 1+ hour of exercise a day, I would say- depending on breed and what sort of exercise it is. You also need to be generally 'around' a lot. I leave my dog for a couple of ours most days. Most people say if you need to leave them for more than 4 hours most or every day then it's not going to work well. They are highly social animals.
Dogs also need mental stimulation. How much, again, depends on the dog and breed. But play time, essentially (including training and playing with interactive food toys and such).
More general constraints on you which people don't always tend to think about include having to find someone to look after a dog whilst you go on holiday, or paying out for kennels (although many dogs will not do well in kennels so costly home boarding may be necessary). Day trips and such will either need to involve the dog or again you'll have to find someone to look after the dog.
Nothing you've said sounds like you aren't suitable. Your daughter is a good age. You'll be living somewhere quite rural. You will be at home a lot. Main concern would be that you don't seem too keen- but then maybe you're just being sensible?
Breed wise consider that sheep may be on the hills. So you'll either need to keep a dog on lead or (if a very safe distance and with the right dog) have a dog who is not likely to give chase to sheep.
Staffordshire Bull Terriers are known to be good with children and are great running companions. My dog and previous dog were SBTs and both loved going out on long runs. Training for up to marathons. The stockier shorter nosed of the breed might not be so suitable, but these days lots of dogs (in rescue, by the way, don't buy a SBT from a breeder) are leggier. Because there are so many of them it's also possible to get a puppy with relative ease from rescue. I always recommend rescuing, we have a massive overpopulation of dogs in this country, but for some breeds it will be difficult (but not for staffies!).
They may be a problem with the sheep, though. They were originally bred to take bait bulls.
Many of the gun dog breeds would probably suit you and your family. Dogs from working lines may prove to be generally quite mad, though, so avoid unless you can provide sufficient mental and physical exercise (i.e. jogging/walks plus games plus training plus agility/scentwork).
They are generally some of the most trainable dogs so would be good fun for your daughter and the family to get involved in early clicker training and tricks and such.
Apart from the SBT I would probably opt for a gundog over a terrier with youngish children.