Another notable difference between the 2005 (and onward) series and the 1963-89 era is the episodic structure. Most episodes today are more or less self contained with about 45 minutes running time, a lesser number are "two parters" the first ending upon a "cliff-hanger".
The original series stories usually spanned an average of 4 to 6 episodes each running roughly 22 minutes. There were exceptions, of course. A few stories ran just two episodes, others were 7 to 10, especially during the first two Doctors' eras. Ecah episode ended upon a cliff-hanger. Think of them as the television equivalent to the the old Republic theater serials like Commando Cody, or even the earlier ones like Flash Gordon or Buck Rogers from the 30s.
Some people, when they watch one of these in "omnibus" format (back to back in a single setting), may find the material bit repetitive, but keep in mind these segments were originally aired at weekly intervals, and at a time before the convenience of domestic video playback technology, they were scripted to recap plot elements the viewer may have forgotten from a week earlier.
Speaking of cliff-hangers", during the earliest years, even stories led directly into the next. Example, at the end of the fourth broadcast when the Doctor and the rest have escaped the ...bleep...to the safety of the TARDIS, they materialize upon a different planet and time and decide to explore the surroundings. As they exit the craft, a radiation metter upon the console climbs into the danger zone as the scene fades to black. In later years, those last episodes of each 'story" would conclude with a bit more finality.
Just something to keep in mind when you finally start watching the "classics". Personally, I LOVE them. I started with the fourth Doctor (as did many other US viewers when various PBS stations started airing the show in the early 80s), so that's my Who. Just remember they are a product of their time and how television in general was produced back then.