Time Lords were commonly encountered in Old Who, and the Doctor's relationship with them was usually antagonistic.
The earliest seasons of Doctor Who in the mid '60s weren't very clear on who or what the Doctor was, and we didn't meet anyone else who called himself a Time Lord (though we met folks "from the same place") until the end of the Second Doctor's era. By the Fourth Doctor's stories the Time Lords were firmly established. They appeared semi-regularly, and more than one older character was retroactively declared to have been associated with Gallifrey in some way.
The Doctor ran into the Time Lords a lot more than he'd have liked (his backstory is that he ran away because Time Lord law is non-interventionist and the Doctor wanted to explore the universe and help people). They often manipulated him into helping them against his will, and sometimes he had to help them against a common foe.
Time Lord society was more often painted as the villain than an ally, but it was complicated: the Doctor did have friends from the old days and one Time Lord was even a companion on his travels.
"The Last of the Time Lords" is an invention of New Who.
Until New Who in 2005, the Doctor was a renegade Time Lord, and there were hints in Seven's era that he might be unusual or special, but he wasn't ever called The Last of the Time Lords or anything close to that. He was unusual, perhaps unique, but not in that way.
The title "Last of the Time Lords," as well as the Last Great Time War itself, is a purely New Who idea. (The Eighth Doctor's adventures in novel and audio stories featured a Time War, but Russell T Davies has said it is not the Last Great Time War. Speculation remains that they are somehow identical or related, but this is --at the moment-- only speculation. At best, the most we can say for sure is that the Eighth Doctor's War probably inspired RTD to invent the LGTW.)