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.
Try telling that to David Tennant. His favourite Doctor is Peter Davison. That is why they had Davison in Time Crash. Tennant wanted to act with his favourite Doctor.
Don't agree with you on saying Troughton is better than Pertwee. I have seen Troughton in The Three Doctors, The Five Doctors and The Two Doctors and wasn't that keen on him. Pertwee is a far better Doctor.
The Troughton-era stories: The Mind Robber, and War Games are really good. I find Pertwee too action-oriented. (I did really like Carnival of Monsters though.)
Now that you mention it, I can see a Tennant / Davison connection. Sort of like Tennant took Davison's version and perfected it. (I grew up on Baker, and he'll always be the default Doctor for me, but Tennant was VERY good.)
From what I've been told, you can start with any of the series since they don't really have anything to do with each other. The 2005 series, like suggested, is supposed to be pretty good. Just start with episode 1 of one of the series and go from there.
I think a lot of people start with the beginning of the 2005 series. But it's really cheesy and I think it takes a while to get to any episodes with enemies that don't seem ridiculous, so I'd recommend holding off judgement for a while. Personally I didn't understand why people liked it at first, but by the end of the season I had gotten into it. (though I've since lost interest)