How do you price a trip to the vet ?

That's a good question. I haven't faced that situation yet, but I'd approach it one of a few ways.
First, the national mileage reimbursement rate set by the IRS is 54.5 cents per mile. That's almost $15 just in mileage! If I felt that was too high, I'd calculate based on the cost of fuel for my car. At about $2.20/gallon and 25 miles per gallon, that's about 9 cents per mile, so the total in mileage would be about $2.50.
I'd probably determine the rate based on time spent and compare it to other services I offer. I offer drop-in visits at $12/visit, and between driving to/from the place and the 30-minute visit, they're probably about an hour. With transport to/from a vet appointment, AND I'm waiting at the vet (instead of going to do a drop-in or dog walk), I'd probably call it equivalent to 2 drop-in visits plus mileage. You could also calculate based on the amount of money you could be making if you had that time free, but I'm not very busy on Rover, so that wouldn't make as much sense for me, personally.
Basically, for me it'd be $24 plus whatever I wanted mileage to be. If $24 didn't make the effort feel worth it to me, I'd do the higher mileage price, which would bring it to $39. (Then I would reconsider because that's more than my house sitting fee, which covers an entire 24 hour period! Granted, there's less driving involved.) If $24 felt like a good rate for what I was providing, I'd just keep it at $24 or add on the lower mileage rate for a total of $26.50. It all depends on how much effort you anticipate expending and what rate would make it worth that amount of effort, then justifying that rate based on your other service rates. You don't have to justify it to anyone of course; you could just say "$30 sounds good." I like to have a basis for what I'm charging to make sure my rates make sense...both to me and my clients. I hope there was something helpful in my thinking aloud. Good luck!