I know it's pretty self evident, you perform actions related to said skill to raise them. But most of them still raise very very slowly. Also, my companions don't seem to raise skills much at all. I've had a surgeon for like the last 15-20 hours of play (set to be clan surgeon) and her skill level has went from 60 to 61. That's all. All in all, the very early game where you have very limited options lasts too long and isn't particularly fun IMO. It just drags on. It doesn't get very enjoyable for me until you have at least a chance to siege a castle or actually contribute to army battles and so forth.
I think just about every gain the player games is way too slow other than making money which comes in a fair rate if you actively try.
When I set myself to surgeon, it raises a point or two a night.