A bit of a warning - you only get what you pay for. The cheap places will do what mortartube describes. The best will do a much better conversion, more 'direct' than just filming it.
I had some done at a cheap place, and they were rubbish - all bright in the middle with the edges dark. I had them redone, and paid more. It was worth it as like yours, mine were very old family films, almost falling to bits now.
So, my recommended company would be
this lot. Reasonable prices, high quality and very rapid turnaround.
I wouldn't bother with videos now, they're nearly obsolete, and prone to damage and wear.