My dog runs on trails and if I get a snake would my dog know not to go up to snakes on the trails?

It really depends on the breed. You might want to try 'snake training' your dog. Your dog may be intelligent enough to learn the difference between your pet snake and a wild snake but I'd be more worried that your dog is going to attack any snake, be it your pet or one on the trails.

Most breeds of dogs at some point in the past had the instinct to hunt and kill promoted and bred more firmly into them, and even if not it is instinctive in all breeds to some level. The level of control the instinct has over the dog versus your control over the dog through proper training would determine whether it approaches a wild snake or not, or attacks your pet snake. If you have full confidence that when you tell the dog 'Stop' (or similar commands) it will stop ANY action and return to you then you make have a chance. However that is not a guarantee for either your dog's health or any pet snake you may get.

Possible Outcomes:
1) Your dog attacks your snake and any it comes across with no bias.
2) Your dog attacks only your snake because a wild snake may be more threatening and scare it off from attacking, but one raised in captivity is seen as an easy kill (though snakes are not domesticated, they are hardly even tame in some cases).
3) Your dog is a perfect angel and with proper training it will not approach any snake.
4) Your snake (depending on type and eventual size) injures/kills/eats your dog.

Personally I say that certain types of pets should never be kept together as a rule.
Take my situation for instance. For years I owned Fancy Rats, I loved them the little buggers. Then I got my first dog (after last of rats died) a Miniature Fox Terrier, one of the more tenacious breeds of terriers and an excellent ratter. I have not even entertained the notion of ever getting another pet rat while I have my dog because it would only end in tragedy, like 3 out of 4 of the possible outcomes I foresee for you. BTW it took me a long time to get a dog and eventually I just smuggled him into the house, ignoring that my parents had forbidden me from ever owning a dog, so I'm just thankful to have my dog and believe every dog owner should be too.

If you really want a snake wait 'til you're older and more mature (aka. moved out of home) so you're Mum has no say in your life anymore. Get a licence and get a snake and by then you probably won't have a dog to cause worry.