In case this hasn’t been settled for you yet, here’s my take.
Got into Who just last year and a couple months. Started with 2005 and watched them all back to back to back on netflix, up through season 4. Then watched the Matt Smith season 5 episodes, and then 6 as it came out on TV.
The first season, with Eccleston, is certainly not the best but I would highly recommend not skipping it. David Tennant’s seasons are the absolute best. Matt Smith is awesome too but you’ll be missing out so much phenomenal TV if you don’t watch the Tennant episodes that I will cry for you over it. Watch 'em all back to back, as many episodes as you can manage to cram in a day. That’s the best way to watch Who (or any TV show I guess).
Then go back and watch some Tom Baker episodes. I’ve seen a few of his arcs and they are enjoyable and fun and interesting.
I tried watching some of the first doctor but I couldn’t take the acting.
But seriously. Doctor Who is some of the best television you’ll ever see in your life. Not every single episode of course, but overall, I came away feeling like I had experienced something uniquely phenomenal when I got through the first four seasons on Netflix.