The first thing I do when potty training a child ( I've trained three and working on a new one two days a week. ) is spend the day in a bathroom or kitchen with no pants on. We have lots of juice, coloring books, reading bookds, and a potty chair. When they start to pee I put them on the potty real fast. They sit there until I think they are done. After a few times they get it. Then we wear big girl panties that I let them pick out. (I haven't trained a boy yet) I get have carpet cleaner on hand for the first week. A few accidents will happen. I use treats and incentives every time they go successfully. Chocolate chips if they go work nicely. No chips if they don't go. Just sitting on the potty doesn't count. Poo on the potty is harder. I have to watch them like hawks and look for the signs of straining and so forth. I have a potty chair always nearby. Big chears and so forth for success.
And no pull-ups do not work. They are just glorified diapers. They are good for trips to the store or outings where you are afraid they might have an accident, but they have to know that they can't go in them. I never put one on a child unless we are going somewhere for a long period of time. Many times I'll even carry a potty chair in the van and stop if my little one needs to go.
At any rate, once I feel they have figured it out then they are not allowed to wear diapers again and if they have accidents, they have to clean themselves up. I'll put my little one in the tub with wipes and tell them they made the mess they have to clean up. I hold the bag for the wipes and help them get it all off but they have to do most of the work.
I've trained both my daughters at 2 1/2 in about a week. My neighbors daughter at 19 months took longer but she may have been a bit too young. She turned two in May and she's done very well. One accident in the last month.
The other one I'm working with will go at my house on the potty but she's giving her parents a hard time. Not sure what to do about that though.