I just recently started kickboxing/Muay Thai and my round to the bag causes a great deal of pain after a while. (20-30 kicks)
I know you build up strength in your shins overtime, and most people advise against rolling bottles or wood on your shins because uneven rolling can cause painful "bumps".
What is the best way to increase shin strength?
It sounds like the problem isn't strength so much as your shins hurting when you hit the bag. The pain will get better if you keep hitting the bag several times a week to the point of discomfort or mild pain, but stopping before severe soreness or severe pain. You'll get used to the feeling. If you only go to class once or twice a week, an additional home workout or two is called for.
I'd make sure to do heavy barbell squats, deadlifts, lunges, and farmer's walks in order to increase bone density, strength, and overall resistance to injury. That will help make sure that the shin pain doesn't develop into real injury. Once those are in place, I suppose sprints, jumps, and other forms of high-impact supplemental training could help.
Yes kicking frequently on a target like a bag or a mitts will harden and make your shins harder. Just keep doing it frequent (at least 2 times a week, more would be better). A more extreme training method is to kick on car tires (some muay thai fighters are actually doing it). Yes, and good leg conditioning (running, jumping, weights, etc) in general will make your legs more resistable.
When sparring and even when wearing shin pads on you can get some big bruises. Sometimes they are so big that you cannot properly train the next day because it hurts so much at the point you're making contact. Then the proper thing is to squeeze out the bruises with your thumbs or fingers or with a bottle so that your shin becomes all smooth again without any bumps.
