The new series was made specifically to be accessible to people who never saw the old series. Even when introducing elements from the classic series, they were explained. Plus, it's modernized to appeal to today's audiences. It will help you gain the appreciation for the concepts in the show, which you will need if you ever watch the old series.
Due to the classic series' low budget, sometimes you need to watch it with a grain of salt. That's not to say it isn't enjoyable--it really is--but some of it had not aged well.
not really. my mum grew up with the old doctor who series and what i did not understand, i asked her about it. i've given a couple of old episodes a try and i quite like the first 3 doctors. i'm planning in giving some more of the old doctors i try soon too.
If you don't want to hit a story-arc episode without any context, you should probably start at the beginning of the new series with Eccleston. They introduce old characters/villains in, but I think they did a nice job of not assuming too much about the viewer's prior knowledge. I really liked the classic Doctor Who, but I enjoyed the new Doctors as well. I don't think it's entirely necessary for you to watch the classic episodes if you don't think you'd enjoy them, most of the older episodes were more tuned to a monster-of-the-week style with recurring villains here and there.