The PRV you show has unions, not pipe threaded joints. Those joints do NOT need tape or dope. The union nut brings two flat ends together onto a gasket or O-ring, normally. The seal is NOT made by the threads. If that matches the one you have (exactly) you will end up with two union ends left over. Generally, if the match is exact, the two new union ends, which means you don't have to mess with pipe threads (or soldering, depending on the pipe type around your PRV as it sits now. That appears to be soldered copper pipes that have been painted.)
The detailed specs indicate that the particular one you linked does have female pipe threaded union ends. If the match of new to old is NOT exact and your pipes are threaded, then you'll need sealant on the new union ends where they join to the pipes.
Pipe threads for water need a sealant - teflon tape (properly applied) or pipe dope (a semi-liquid or paste type product) are the options there. Teflon tape is slightly harder for a novice to get right, and the cost of a small tube of dope and a roll of tape are not dissimilar. So I'd suggest dope.
If you choose tape, it must be stretched as you roll it on in the same direction the threads run, and it must not be over the end of the pipe, where shreds can break off and clog valve orifices downstream.
Since it appears that your pipes are soldered copper, your choice of new PRV should be one with solder ends, but truly the easier option is to make sure the new PRV is an exact patch to the old one so you can just unscrew the union nuts, slide out the old PRV, slide in the new PRV and tighten the union nuts. They are designed that way so that they are easy to change once installed the first time.
You can also buy rebuild kits to replace just the working parts of the PRV if the main body of the valve is not damaged.