Two week itinerary help?

Hello. It'll be my first time travelling to Japan in September and I really want to minimize the spending involved.

My itinerary consists of 7 days in Tokyo and then 3 each in Kyoto and Osaka, heading back to Tokyo the final day to get to the airport.

I'm planning on buying a suica card for the trip in Tokyo, but I'm not sure if it'll be worth it to buy the rail pass for the next 7 days. Would it be better to buy individual train tickets to Kyoto and Osaka? I'm travelling with my boyfriend and we agreed that we're not interested in seeing temples, so I'm not even sure if it'd be worth it to see Kyoto and Osaka, both of which are known mainly for temples.

And finally, I'm not quite sure how the suica card works. Do you have to enter your destination or does one swipe at a station grant you access to every stop on that line? If it's the latter case then I have to minimize the number of times I get on the train right? How much is it normally per swipe? Sorry for all the questions, but thanks for the help!

I'm planning on buying a suica card for the trip in Tokyo, but I'm not sure if it'll be worth it to buy the rail pass for the next 7 days.

If you are looking to travel as cheaply as possible, you should not buy a rail pass and use overnight buses for your trips between Tokyo and Kyoto/Osaka:
http://www.japan-guide.com/e/e2363.html

I'm travelling with my boyfriend and we agreed that we're not interested in seeing temples, so I'm not even sure if it'd be worth it to see Kyoto and Osaka, both of which are known mainly for temples.

No, Osaka is not known for temples. You could save a lot of money by dropping Kyoto and Osaka. I recommend to do some better research before making a decision.

And finally, I'm not quite sure how the suica card works. Do you have to enter your destination or does one swipe at a station grant you access to every stop on that line?

You just swipe the card at the starting and end stations. It is explained here:
http://www.japan-guide.com/e/e2359_003.html

How much is it normally per swipe?

It depends on the distance traveled. Fares start around 120-160 yen for very short distances. You can look up train fares here:
http://www.hyperdia.com/