I always start with the electrolytics. After they're done, you can power up the set without much anxiety, and the performance of everything else depends on a healthy power supply.
As Dan said, doing one at a time will build confidence. You can power up the TV briefly after each replacement. If something suddenly gets a lot worse, you have only one or two caps to double-check, not two or three dozen.
I check off each cap on the schematic and the parts list when I replace it. This is another way to confirm that you have the right capacitance value, etc., in the right place.
It's not rocket science, and you don't have to do it all at once. I'm a terrible slowpoke compared to some folks. Lots of distractions & interruptions around here. If you find yourself getting frustrated, take a break. The TV will still be there the next day.
This article has more general info about recapping:
http://antiqueradio.org/recap.htm .
Phil Nelson