TV comes on and then cuts out

Sounds like a defective TV to me. Nothing wrong with the remote if it does that much.

Changed batteries in remote. TV comes on using power button on TV, clicks, then cuts out.

It's not your remote. There are mnay things inside that could cause this.

Taking the back off and blowing any dust/dirt out is a first step. Double-checking for any loose wires is next although you have an HDTV so it will be harder.

Try again.

If not then I would look into contacting the manufacturer (if its in warranty). As I stated above, there are many things inside that could cause this.

The remote won't turn it on and back off immediately... its a defect inside your TV. While I may understand the causes ( I have a degree in Electronic Engineering and know how tvs work even if I am working in a different field) they are not something someone without training can service. And in fact there are voltages inside a TV that can kill you. Without schematics... the correct test equipment and the training to know how electronics work... all you are going to see is wires and lumps of difference shapes sizes and colors with wires on them. You will have to return this if under warranty , take it to a repair shop if it isn't... or throw it out and buy a new one if you can't afford that.

However, blowing the dust/dirt out is something you can try. I've seen this before and it was just a dusty TV (very, very dusty)

When dealing with any sort of issue. You want to start with the easiest solution as they are usually the cheapest/free and can usually be done by the individual.