I'm in debate on whether or not this questions belongs on this site, however I feel like it's an honest query, and when in doubt, I think our default response should be to be answer the question as opposed to down voting, or saying it is off topic, without suggestion an appropriate forum in which to repost.
TBA - There are a number of things you can do to "speed up" web applications, however the majority of them need to be done on the host end, although with advent of extensions individual apps may have options that will allow you to decrease load on your system, and increase performance.
Depending on the web-app they may have specific plug'ins for the browser that will speed things up, but that would be on a case by case basis. A few examples:
http://imo.im has a chrome plugin that allows you to use the web-app even if it isn't open.
Digg.com has extensions and smart bookmarks that allow you to use the functionality of the site without visiting the site and loading the pages into your computers memory.
Even Google has a toolbar which will keep you up to date on whether or not you have new messages, so you don't need to keep Gmail loaded into memory.
Check for Chrome Extensions: https://chrome.google.com/extensions/?hl=en
Check for Firefox Extensions: https://addons.mozilla.org/en-US/firefox/?browse=featured
You mentioned Google Gears - which I believe is being "dumped". Besides the obvious "speed up your computer" options, your only universal option for speeding up web applications is to get a better/faster/more stable internet connection. Using a better or different browser may speed things up depending upon the application, but that would vary by app for the most part.
From the host side, there are a number of ways to speed up a web application, entire forums, books, blogs, sites have been dedicated to the subject. A few notable options that the host can choose to implement are: Caching, Optimizing code, Optimizing data calls/data structure, load balancing accross multiple servers or improving the host hardware (servers, etc), even things like switching away from "heavy" plug-ins such as flash would benefit their users experience.