It may not be that your daughter doesn't understand taking her panties off... she just hasn't mastered that skill yet-and in all the excitement of going potty, isn't able to work on it at that moment. An important aspect of toilet training is the removal of pants. Many experts that I've read state that being able to take off clothes (specifically the lower half: pant/skirt) is a requirement for toilet training. Otherwise, how can they actually go potty in the toilet? Look at this link for that and other cues to know the child is ready: http://www.askdrsears.com/topics/child-rearing-and-development/toilet-training/6-steps-d-day-diaper-free-day
I would suggest playing some games with her to help her want to learn to take off her clothes, if that's what you really want her to do right now-otherwise, take it easy and wait until she is ready in that way, as well. It won't be long. Both of my girls were ready and wanting to between 22-30 months...although many kids aren't ready until closer to 3.
Also, in this article: http://www.askdrsears.com/content/potty-training it suggests emptying the diaper/underwear into the toilet, when there's a poop accident. We did this with both of our kids saying, "Let's put the poop in the toilet where it goes/belongs." Helps them see that even with an accident, the poop still belongs in the toilet, not for Mommy/Daddy to clean up.
All of that being said, my 4yo still doesn't always like to take off her pants to go potty-no accidents, she just waits for someone else to do it. She is entirely capable of removing all of her clothing as she does it several times a day-there's just something about the bathroom that makes her not always want to. However, she has taken great strides in this and takes care of things herself more often than she used to. It will happen.
We didn't go directly from diaper to underwear-we went to pull-ups (which didn't work for either of our kids) to cloth training pants to underwear. There are some great cloth training pants on cloth diaper sites that I found to be more effective than the thinner Gerber ones I could find locally. They hold one pee without leaking (for the most part) but make them feel the wet, as opposed to diapers that keep them dry even when the diaper is wet..and our girls never noticed the cool core in the pull-ups (when they got wet) it didn't bother them in the least.
Here is an article discussing pros and cons of disposable training pants and underwear in toilet training: http://www.livestrong.com/article/113065-disposable-training-pants-vs.-underwear/