It takes some practice for kids to really learn what it feels like to need to use the potty. When we first tried potty training our son, we put him directly into underwear and it was a big flop. We think the underwear just felt too similar to wearing a diaper to him.
So the second time we had a go at potty training, we let him run around naked from the waist down for a long weekend (3 days) at home. He had a few accidents (mostly pee), but seeing the pee run down his leg helped him make the connection between the sensation of needing to pee and actually peeing. We kept the potty in the living room so he wouldn't have to run far if he needed to pee, and every time he got something in the potty we made a big deal about it (we sang, we danced, he got stickers....) even if it was only a little bit and the rest got on the carpet. We kept him plenty hydrated, and asked him very frequently if he needed to potty. If we felt like he hadn't used the potty in awhile, we'd make him sit on the potty for a few minutes either until he went or it was clear he didn't need to use the potty (4-5 minutes or so--we'd read a short book while he sat on the potty). After a few hours of this, he started getting the hang of it and would run to the potty if he had the slightest inkling that he needed to pee.
My husband found this method on the website of a woman who has supposedly potty trained children as young as 1 year old. She recommends that you continue to have your child naked from the waist-down whenever they're at home for 3 months after the initial weekend of training, that you have them use the potty before you leave the house and immediately after you return home so that they get in the habit, etc. We circumvented some of the later suggestions because our son was older and didn't really need the constant reinforcement that younger children need. Within a couple of weeks he was consistently using the potty and we'd only have the random accident (usually when he was so absorbed in what he was doing that he wasn't paying attention), but that more or less too care of itself after a little while, too.