Congrats on your new baby and welcome to YT! It great that you put so much thought into your new puppy parenthood!
A Yorkie-poo is a cross breed between a Yorkie and a Poodle...
Your little one will probably sleep most of the time on your way home. I would drive straight through but with a pit stop every hour...
Water on a walk is only necessary if you are out more than one hour or if it is really really hot (and then a walk might not be a good idea anyway...)
Until your puppy had the second round of shots I would not take it anywhere. After that take it easy until he/she is about 6 months.
A word to more vaccination: It is a fairly new concern not to over vaccinante toy breeds. Maybe you could do some research on that too.
What your baby needs most is love, love, love...