You could go the professional route with several vendors, most of which are in the Los Angeles area, and they'll clean the films, splice them together, and use the same telecine equipment used to transfer feature films.
My grandmother, however, took her reels to Costco. They transferred everything to DVD for her for something like $20.
Yeah, most places have someone that will do it professionally (just look up Video Production in the yellow pages) as that is how a lot of video companies make extra money, especially around the holidays.
Costco is another option as well. Costco and most the other professional places with run some software on the digital files to clean it up/stablize and shaking. Again, you can find similar software to do this your self.
The big thing comes down to price. Costco for example is $17.99 for the first 150 feet, $.11 per foot per additional foot.
150 feet is just over ten minutes of film. If you have an hour of film that's going to cost you a little less than $100. Which is a good deal for all you get.
The other advantage of doing it yourself though is the ability edit just the parts you want, add text and transitions, music, etc. All features like that are going to bump up the cost into the $100s from a professional.
