Not all dogs like crates, not all dogs will be destructive and not all dogs are hyped up in the car. Get to know your dog, teach it to leave things it isnt supose to chew and give it things to chew that it knows it can and encourage it to chew on those things.
Good luck

I would get a crate so that she can adjust to being in one. She'll have to tolerate one when she goes to the vet, and as you mentioned, it will come in handy when you visit relatives.
Start her off by leaving the door open and making it a place she can receive snacks and take a nap. I would also use it as a place for her to sleep at night. Where did she sleep last night and how old is she?
Until you know she can be trusted in the house, especially potty wise, I'd either keep her in a small room, crate her, or take her with you when you leave the house.

We got a gently used second hand dog (7 yrs old) two years ago, and I don't think she had ever been crate trained. We use baby gates to limit her access to parts of the house we wish to remain fur free. We close the gates over night, and if we will be out of the house for 5 or more hours during the day.

Well behaved or not, a mix of beagle and rat terrier could get in a lot of mischief left unattended. Every dog is different, but I'd err on the side of crate until you are certain she will be OK with free run of the house when you're gone.

You also have to expect an adjustment/learning period with a new dog. Grown or not, they don't know the "rules" in their new environment. Easier to establish them from the get-go than to try and break bad habits after the fact.

The beagle I adopted was a hellion. The cattle dog mix I adopted was perfectly mannered. You never know :)

I've never used crates for my dogs but if I go out for 20 minutes then I just let them have run of the house. If I'm going out for a few hours I put them in the sunroom and they also have access to the laundry, since both rooms have hard flooring so if there are any accidents it's easy to clean up and have a quick mop around. I basically just don't want to be trying to clean up messes off carpet etc. but it's really only been an issue when my dogs are old and can't hold it as well. Since this is a new dog I'd just pick a room or two that you can close off until you get the housetraining sorted (even a housetrained dog will need a bit of retraining in a new home to learn the new toilet routine) and you can use that area to confine her when you leave the house instead of using a crate. put a nice comfy bed in there and a water dish etc.

There are usually lots of used crates on Craigslist, so no need to put a lot of money in a new one if you end up only needing it for a short period.

As weed mentioned, you have an interesting mix there!
Beagles can be difficult to housetrain and terriers........well, they're terriers! :)
Each dog is so different and the characteristics of each breed isn't set in stone, so......congrats on adopting from a shelter and hope you continue to enjoy your new addition to your family!

Totally dependent on the dog. I had never used crates until our current two dogs. It was necessary for both, but in totally different ways. The big dog needed to have extreme aggression dealt with and using an open crate was part of that training. Once the aggression was modified, there was no reason for the crate.

The small dog needed the security at first. I put the crate by my side of the bed and made a cave by covering the top. Once she was house broken, and her confidence boosted, the crate is left open so she can use it as she pleases.

However, if the small dog is not exercised fairly regularly, she can get a bit destructive---tissue boxes are her favorite. She will get into the living room trash during those times---but not the kitchen trash.

Dogs are often destructive because they are bored. The more energetic the dog, the more easily it can get bored. Crating can be a good part of a dogs life if it keeps the dog from being destructive. But it is a very bad idea to take the place of exercise.

Crating in a car is much better than allowing an energetic, hyper, or scared dog loose and create distractions for the driver.

Crates are my friend. I never had them for my previous dogs but when I adopted a Boston mix four years ago, he came with a nice wire crate which he loved. So when I adopted my dachshund two years ago, I put him in a plastic crate until I could order the exact crate as the other dog. He was around 3 or 4 then and was not going in the crate. We put him in since we were leaving for the evening and have four indoor cats and had just gotten him. His new crate came two days later and both my dogs love them. In fact although we're home all the time the crate doors are open and many times they will be in one of the others crates. Sometimes there's a cat in there too. I have nice fluffy blankets and a down pillow on the bottom. Their crates are their sanctuary and they are only locked in at night. .

lily makes a good point. Some dogs do much better in a wire vs plastic crate. A wire crate is my preferred choice for at home use and car travel.

I adopted Artie a small 2 year old terrier mix in December - he was already crate trained. I've never crate-trained any of my previous dogs. I really like it. Artie is a sweet dog but very excitable in certain situations - strange men, cars/trucks, vacuums, evil squirrels & birds etc.. He calms right down in the crate - and in fact goes in voluntarily when bored or scared.

I have a Jack Russel and a Beagle. I have never used a crate and they are both extremely well behaved.
They have NEVER gotten into the trash or done anything at all destructive in the house.
I leave them at home for several hours at a time and nothing happens except a lot of sleeping.

However, I think their good behavior is due to my being home all day and readily accessible to their needs so they have no reason or desire to act out in boredom or anxiety.

Okay, confession, once, my JR pulled a piece of paper towel that was hanging over the edge of the inside garbage can and shredded it. I had drained bacon on it.

I feel so blessed with them but I think so many dog owners aren't able to give their dogs what they need and then get bad behaviors that are attributed to being "just dogs".
Not so.

Before we got our puppy, I was totally against crates because I thought they were cruel. I still do when people leave their dogs in them for hours on end. However, we had a very large, over sized carrier in the basement we had used to transport our cats across country in and decided to bring it up so she could sleep in it. She took to it immediately and even as a new puppy never cried once when she was in it. We never punish her with it, it's her "room" to spend time in and for sleeping. She knows it's her space and loves it; in fact, she's taking a siesta in it as I write this. :c)

I'd highly recommend using one now. If for no other reason it could help the dog feel more secure in his/her new surroundings