As to how long it takes you to learn - it depends on how much time you have to practice. I play the viola casually and I've played for about... huh, 15 years or so, but at the height of my ability I was only practising for an hour a day at
most, whereas someone who's really dedicated will practise for longer. Someone who's not used to longer practice hours and doesn't work up to it can actually injure themselves quite badly (typically the left hand, rather than the bowing hand).
When you start out you have to learn to maintain the instrument (tuning, putting resin on the bowhairs, tightening the bow before use and loosening after, cleaning the strings and the body, etc.), how to hold the instrument and bow, and basic bowing (just playing the different strings without putting fingers on the fingerboard). Your teacher will typically tune your instrument for you for the first few lessons (more if you're a kid) before you get taught how. Holding the instrument and just playing the strings without fingering gets picked up in one or two lessons, though your teacher will generally keep correcting you for a while after you move onto more advanced things. Then you can start working on simple fingering (generally on one string, without changing). Twinkle, twinkle, little star is pretty much a staple for this.
Then you start working on scales and arpeggios and can move onto more advanced pieces, and I don't remember how quickly that progresses - it's been a while
. Note that all of the previous stuff can be taught without knowing any musicianship or how to read music, but soon after that point you have to teach that as well.
On the other hand - string group pieces often tend to have a very simple viola part, so someone could start viola and work up to learning that part in a couple of months as long as they had a decent grasp of rhythm, and regular practice and lessons. Depends how advanced the piece is, and I know you can get simple ones. To get playing the really advanced stuff takes years and years though.
Last thing worth mentioning - anyone who can play the violin can play the viola, and vice versa, though they can get thrown by two major things: 1) the relative weight of the instruments, as there is no standard 'full size' for a viola, and is mostly based on your height (typical range from 15 inches to 17 inches for the size of the body I think; I'm not that tall, and mine is 16 inches) and 2) the relative finger spacing, as the viola is quite a bit larger and thus fingers have to be spaced a bit wider apart (this is also a result of the viola being a fifth lower).
