We potty trained my 2 year old little boy over 4th of July weekend. He woke up that morning and I told him that he was going to be a big boy now and not wear BABY diapers anymore. He was so excited. I told him that it was his responsibility to tell me when he had to potty and if he had an accident HE had to clean it up. The first two hours he had 4 accidents. He thought it was fun cleaning it up the first 2 times, then it wasn't fun anymore. I never punished him or spanked him for wetting his pants, I just handed him a paper towel and we cleaned it up together. On day two, we went to town and he didnt have an accident. He had a few accidents over the next few weeks, when he was playing and not paying attention, but after that he did great. At night we did put him in a pull up/diaper...but we told him they were "night night underwear" and that you can't pee pee in those. He did have accidents in those for a while, but, that's to be expected.
Just remember to stick to your plan...it might take a few days, but eventually they get the hang of it and it's sooooooo nice! Just dont give up!
Now pottying in public restrooms...that's not too fun! ;) GOOD LUCK! :)