Spray soapy water around the bead on the rim and see if there are bubbles. If there aren't any spray the whole tire and valve stem and look for bubbles. This is the only way you are going to know for sure. Right now you are just speculating where the leak is.
You need to get some soapy water and see where the air is leaking. Obviously it is not obviously the rim. IF it is the rim, the rim might be able to be fixed if there is a bur on it or some other blemish. It would depend a lot upon what your wheels are made of and if they are coated with anything.
It will cost you to have your rims wire wheeled to seal better. Some rims have been known to have pores in the metal leak. I have seen chrome wheels peel at the bead sealing area and never again hold air. Even with bead sealer which is a sticky mess. New rims are sometimes necessary. Remember no chrome.
Take it back to where you had the tires replaced and they can figure out what the problem is. It is not necessarily a bad rim. The air valve could be loose inside or they did not clean the rim properly.
If I owned new aluminum wheels I would want to know exactly where the air leak is. They are easy to find with 75% dish soap and 25% water in a spray bottle and the wheel off the car.
The valve might be the faulty one, and also do check for any sharp thing that might be sticking in the tire of your car. if the air leaks out within a week or so then i think it is the tire.
Dittos to others answers about cleaning the sealing service when tires are installed. Other problem: some aluminum/mag wheels are pourous and with age can leak right thru the metal.
They did not clean the bead on the wheel rim when they replaced the tyre. It needs taking back and doing properly. Simple as that.
No you do not need new rims. Go back to the tyre dealer and get it removed and refitted correctly. If the rim is not damaged then no need for a new one.
If it is an allow they do go porous. I cheated with mine and had Ultraseal put in. No issues any more. :-)
I needed new tires anyway so I had them and the valve replaced so I know it's the rim, air slowly leaks out after about a week.
The rim doesn't look bent or rusted at all around the edge, is buying a new rim the best thing to do?
You could consider putting an inner tube inside the tire.