Nah. Classic Who is fun, by all means give it a chance, but you don’t _need _ to watch a minute of it to enjoy the new version, and Classic is very much not to everyone’s taste.
If you do decide to try it, definitely start with Tom Baker and work outwards from there. It helps to watch each Doctor’s run more or less in order if you can, but it really isn’t very important.
My local PBS station seemed to feel the same way.
But, to the OP, if you haven’t seen Doctor Who, then I don’t know if the classics are a very good place to start. For one, they have some pretty terrible production quality, and that may turn you off. They also assume that you have been watching from the beginning, and expect you to know a lot of lore and characters that you probably won’t.
I grew up watching it, so I found it entertaining at the time, but I wouldn’t recommend it, personally. If, after watching the revival series, you still have interest, want to see more of the Doctor, then it will still be there.
Just start at Season 1 with Ecclestian, watch through Capaldi, and then wait with the rest of us for the next Doctor.