Start with the restart (2004/Christopher Eccleston) - then you can always go back to the old ones if you like, but they were way behind in cinematography in those days - and it shows. (The DVDs are in my library - so check it out, you may not have to pay anything to get caught up.)
David Tennant is probably the best of the current series.
I use our library a lot...well my Wife does for me. It's amazing what we have available for free. We also have a ton of audiobooks that can be "checked out". It's great. Wanted to read "Profit First", but was being honest with myself and knew I wouldn't have time so we looked and sure enough it was there for free.
I'll keep in mind when to start. I have seen screen shots of old episodes and they look very B rate even for the time. So starting with 2004 era is probably a good idea.