We started DD at 18 months. Introduced the potty and let her play with it. Used it as a chair, put toys in it, etc... At 20 months or so, we really started the real training. By two, she was diaper free in the house, out of the house we used pull ups (no difference between that and a diaper in my opnion). By just over two (25 or 26 months) she was diaper free out of the house. She was diaper free at night by 2.5. I also didn't use a smaller toilet seat, just taught DD how to hold on with out falling in. When in public, I held her. I also made sure I knew where all toilets were when out in public. Because when a potty training toddler says "pee pee" or "poo poo" they mean it NOW!
A big part of it is having the vocabulary and understanding to tell you what they need to do. We let DD go half naked around the house or randomly put her on the potty/toilet. Gage when she would need to go by watching how much she drank. Poop was the other issue. Wow.... there were some BIG messes and MANY pairs of underpants thrown out. I think I even threw out a few pairs of pants.