I follow a tried and true rule: Never play a newly purchased cartridge game until you've cleaned it. This goes for new consoles as well.
The best thing I've found for cleaning cartridges is a Magic Eraser cleaning sponge and isopropyl alcohol. The Magic Eraser will absorb the alcohol and acts as a mild abrassive as well.
Since you can't use this method witout opening the cart you'll need a gamebit/security torx/philips bit to open your game and remove the circuit board.
Hold the board and rub the contacts back and forth over an alcohol soaked eraser sponge. Here's a half-cleaned nes cart to show how well this works. try not to get too much alcohol onto the green protective laquer on the board as some varieties are soluable in alcohol. (I haven't run into this problem with a game yet, but I have on other circuit boards)
On carts you can't or don't want to open just use a qtip.
Remove a ciruit board from a cartridge. Cut a strip of regular printer paper to the width on the connector. Fold the paper over the connector edge. put a bit of alcohol on each side of the paper and plug/unplug the game board several times.
I was rather supprised the first time I tried this that the paper survived but it actually works very well.
You can't effectively clean the nes connector without taking it out of the case. Once it's out, use the paper method above to clean it. Pay special attention to the top pins as they are the critical ones for avoiding loading problems.