I doubt a new router is going to solve your problem. You already stated that web browsing is not an issue -- Google Play is just another web site and the 'Play Store' app is simply an app that interacts directly with the Google Play site instead of going through the browser. If your wireless router was giving you online connectivity problems you'd also be having problems using your browser.
Maybe a more likely issue to look at is the 'Play Store' app. Go into your 'Settings' menu, select 'Applications', select 'Manage applications', scroll down and select the 'Google Play Store' app. Click on the 'Force stop' button at the top to quit the app, click on the 'Clear data' button in the middle, and then the 'Clear cache' button at the bottom. Now start up the 'Play Store' app like you normally do and see if there's any difference.
Also, if you go back into that 'Manage Applications' menu, at the bottom will be a 'XXXmb used' figure on the left and a 'XXmb free' amount on the right. I've found if my 'XXmb free' number drops below 20 MBs or so things start acting kind of out of sorts in general and I'll have to clean up various app caches.