Couple things to add - traditional ways to enlarge a slide would be an interneg print or Cibachrome (Ilfochrome). With an interneg, the lab would photograph the slide with special negative film and print that like any other negative. In practice, some colors work better than others and there can be a contrast problem, especially in the highlights. A Ciba is a very high quality print that looks very similar to the original if done correctly. This is almost a lost art, and if you can find anyone who does it they will charge a lot.

Digital scanning and printing has dominated slide printing for a few (several?) years. It's possible to adjust color balance and anything else in Photoshop and the results are excellent. The equipment to do it can be very inexpensive, and even high end digital printing isn't too bad compared to the price of a Ciba (or digital itself a few years ago).

Typically, the slide itself is the finished product. With negative film it's the print that's the final product so darkroom adjustments can happen during printing. What you choose to use can depend on a lot of things - mainly what your client wants if it's professional work. Slides will give you great color and sharpness but don't have the latitude that print film has.