I wish there was a simple "push a reset button" fix for your set, but unfortunately this is not the case .
This is one of RCA's ITC222 Chassis based TV's - Personally I have worked on quite a few, and have never really been a big "fan" of them.
The Power indicator coming on and shutting off 3 times - this tells me one of two things:
1. This set used a wave soldering process, which makes the solder joints around the terminals of the flyback transformer develop "Ring Cracks" - this will cause the problem you are having..Sometimes resoldering the flyback transformer will cure this problem.
2. The Flyback transformer is bad. It will attempt to fire 3 times, and if unsucsessful, it will be turned off by the Processor in the TV. The Flyback is what generates the high voltage to "fire" the tubes and light the screen. The flyback transformer can be replaced, although if you have no previous experience with electronics repair i would not reccomend you try... there are very dangerous voltages involved with this component that could injure or kill you
If you take it to a shop, the transformer will cost about $60.00 and the labor will be around 180.00 or so, depending on the shop -
Now is where you may want to look at the pros and cons - this set originally sold for around $800 - $900 dollars new - I would guess it is probably 6 years or so old now..
The expected life of this set is only about 10 years, so If the amount of the repair is worth having this set work for another 3 years or so, I would say have it fixed.
You could also put the 250.00 towards a new set, and get an extended warranty as well, so you can be trouble free for at least the first few years..It really comes down to your decision.
I hope I have given you the Information you were looking for, and If you need anything else, be sure to let me know!