What kind of car and how old? I'm guessing this is an automatic transmission?
If so, your transmission fluid may be burned or low. Some radiators have a section for cooling the transmission fluid. If you just had the radiator replaced, they may not have put sufficient fluid in to fill the cooling section of the radiator (or maybe they didn't even route the fluid into the radiator). First step: Start the engine on level ground and let it warm up a few minutes. While engine is running check the fluid (this is the procedure for most cars, but check your owner's manual). If fluid is low, some more needs to be added, but very carefully to prevent over-filling and the funnel needs to be absolutely clean. Also, only use the fluid recommended in your manual. DON'T put fluid in the oil fill. While you're looking at the fluid, check the color. If it's light red and clean, it's ok. If it's dark brown or dark red and smells burnt, it's bad fluid and needs to be replaced. Different things will cause the fluid to be burnt, so replacing it may not fix the problem.
Now here's the tricky part. If you take it to a "transmission place" like AAMCO, they will try to sell you a rebuilt transmission no matter what. They will even refuse to replace the fluid, because they know once they do that the transmission will probably run fine and you won't buy a replacement. You must go to someone who is trustworthy. If you need to rev it to get it into reverse, it's because it isn't developing enough pressure to operate the reversing valve. This can be low fluid, plugged lines, trapped air, bad fluid, or a software problem on high-end transmissions.
Maybe you can just park it so it's facing uphill?
If this is a very new car, or a luxury car, it may be an electronic transmission. These are touchy, so only a qualified mechanic can troubleshoot and repair it.