What kind of caulking do I need?

Got a little work done today and boy, was the old stuff hard to remove. It wasn't to some degree, as it pulled off in a string pretty well. But, the full width of it usually didn't pull up, thus leaving some on one side and some on the other side to be scraped off and it wanted to come off in itty bitty pieces for the most part. Sometimes it came off easier with my fingernail than trying to scrape it. I wonder if I'm trying to get the surface TOO clean? A 9 ft. section took over an hour to remove. If I have it where each side of the surface can get fresh caulk, is that sufficient or will any remaining old caulk on the outer edges keep it all from adhering? Also, I may have put the proceal on a little too thick, but on places where it looked like I moved a little too fast and thus a thinner/narrower bead, it looked like it wasn't enough/almost like I was seeing the edge through it. I know it's hard for anyone to say without seeing it. I may try to mask off the borders when I work on it again.