Andrew's guess is right. A surrey was a type of open-sided horse-drawn carriage from the late 19th and early 20th century. If you google-image something like 'surrey carriage' you'll find lots of pictures of them. You'll also see that some of them have decorative fringes around the edges of the covering canopy.
A gig is a more general term for a light horse-drawn carriage, and 'slick' means smart, smooth or stylish. So 'slickest gig' means something like 'smartest carriage'.
Do bear in mind that 'Oklahoma!' was written in the early 1940s and is set way back in 1906, and - needless to say - both transportation and language has changed a lot since then. It's a classic American musical with some great songs to sing along to. Enjoy the music, but don't worry too much about the language - this kind of vocabulary isn't exactly useful in 2016!