It's frustrating to have to get up every few hours to take the puppy out to potty, but it is normal. They have such small bladders at that age, and really don't know how to hold it very well. As they get bigger, they'll start being able to hold it longer and longer.
Some folks recommend taking up the water by 7 or 8pm, in order to minimize the need for overnight potty breaks. Personally, I left the water down all the way up until bed time. I didn't want to deny her water if she was thirsty, just because it was inconvenient for me to wake up in the middle of the night to take her out potty. As it turns out, now that she's older, I'm the one who gets up in the middle night to go to the bathroom.
As for the actual taking the puppy out... the goal is to go immediately to the potty place, say the potty command, praise the potty, and then go immediately back to the crate. No playing or other interaction at all. The dog needs to learn that a potty break is not play time.
If the puppy is still small enough, and you're able, I recommend picking it up... for two reasons. First, less chances of an accident midway. And second, less chance for a playtime distraction.
As for the crate... this is what I did. Whether or not it's the right thing to do, I don't know, but it's what I did, and it worked for Cassie. First thing is to establish oneself as the pack leader. This is just something in general. But as a pack leader, the dog will follow. So, when I would put Cassie in her crate, I would then lie down outside the crate, and close my eyes and pretend to go to sleep. She would see that it was sleepy time, and go to sleep herself. Once she was asleep (usually no more than 2-3 minutes), I'd get up and go to bed myself.
Pretty soon, she would realize that crate = sleepy time. So, I'd say "Crate!" and she would walk into her crate, get a cookie, praise, and then she'd circle, lie down, and snooze away.
Some general tips for crating... It's important that the dog understands that the crate is a good place, a happy place, and a safe place. Never ever put a dog in their crate as a punishment, or with a negative tone. A time out or a calm down period is fine, just as long as they are praised when they get into the crate. As long as they see their crate as a good place to go, and one with a purpose, they'll do just fine.
In regards to travelling in the car... I personally use a crate versus a restraint. Puppies like to chew, and seats look awful tasty. Between claws and teeth, I knew my seats had no chance. I used a small/medium travel crate that I got at Walmart or Petsmart. Once she outgrew it, I got a collapsable travel crate for my car, as a large hard sided crate would not fit into my sedan.
As tempting as it is, I wouldn't take the puppy to any public places until it has had the appropriate shots necessary to protect it. Even just outside for walks, you have to be careful, as the puppy can pick up things from other dogs.
Once all that is done, I highly recommend heading out to PetSmart or whatever local pet store chain you might have that allows pets. It's great socialization for the puppy to see all those people (who by the way will fawn all over the puppy). Just of course be careful of other dogs.