I have recently started a job working in a nightclub/restaurant/bar as security. Having to deal with loud hip hop and dance music all night I want to listen to something soothing during my drive home.
I know nothing of any classical or instrumental music. I do know I like violins though. A few years back I bought the soundtrack to “Pride and Prejudice” and I love this song: Youtube.
I also enjoy the beginning of this song: Youtube video of sand animation.
Try some Ralph Vaughn Williams.
Here, listen to The Lark Ascending and see if you like it.
You can buy whole bunches of relaxing classical music collections, so that’s a place to start. Since you like violins, maybe look into string ensemble music, especially Classical and Romantic eras. Try Haydn, Mendelssohn, Brahms.
By the way, the piece from the Pride & Prejudice soundtrack is based on a famous theme by Henry Purcell, a baroque English composer. It’s from “Abdelazer, or the Moor’s Revenge”.
The tune owes a lot of its fame to Britten’s use of it in the Young Person’s Guide to the Orchestra. (Ignore the comment to the video that says it’s the second part, that’s the opening.)
However, the version of it in the OP’s example, plus the other clip, leads me to suggest that music from a Celtic-crossover genre is going to fit the bill rather than a lot of classical music. (Most classical music should be anything but relaxing 
)
h. sapiens posted good advice. Also look for the piano and orchestral music of Debussy and Ravel. Also Chopin’s piano etudes, nocturnes, and preludes. If you have a CD player, you can probably borrow some CDs from your local library.
I would recommend:
“Pavanne for a Dead Princess” by Maurice Ravel
“Adagio for Strings” (I think this is the correct title) by Samuel Barber
Beethoven’s Ninth Symphony
Shostakovich’s Fifth Symphony
Mozarts string quintets. I have all of them and they are all wonderful.
Regina Carter does jazz treatments of classical pieces. I have her CD Paganini: After a Dream and it is quite good.
You might also check out some jazz. I find Paul Desmond and Glenn Miller Orchestra to be quite soothing. Also, I recommend a CD entitled The Gentle Side of John Coltrane.
You want to listen while driving? Find out if your city or town has a classical music radio station. Big cities do, some smaller cities may. NPR has some classical shows (at least where I live).
Listen to that station. When something you like is played, listen to see what it was. Remember the composer, and look for other music by that composer later.
Radio is great for exploring music – you will definitely hear things you wouldn’t hear otherwise.