While I'd like to start with the most important room, the console room, I tend to get stuck on that one. So, have the next important one: the lab.
The Doctor's lab/workshop/toyroom... whatever you'd like to call it, is a place of peace for her. Or is, most of the time, anyway. This is where the magic happens. Sexy science magic, where thingamagigs and whatchucallsits are made. Where she's found cures for zombie diseases and has created worldwide water filtration systems and edible plants that can survive years-long droughts.
Where her newest version of K9, codename Hendrix, is being born.
The room is made up of a hodgepodge of technology across different eras, some from Earth, some from other places. Computers are slapped together in strange and interesting ways that only The Doctor seems to understand- it's a wonder any of them work at all, and boy, do they work, as she uses them to hack into even the most sophisticated systems when she needs to. Or is bored.
While she has a normal kitchen (there is, in fact, more than one small kitchen on the TARDIS, but more of that later), the one here is more than adequate for her needs. Given the bathroom and sleeping area, where often times she stumbles through the sliding door to collapse onto one of the beds, there have been times when she hasn't left the lab for months days at a time, especially as the fridge possesses a food replicator.
If you're of the scientific mind, or are simply just curious, she'll more than welcome you in as long as you don't run amok. There are things that are breakable, dangerous substances, and more than likely, planet-ending doohickies.
Don't worry- those last bits are safely tucked away. Probably.