I've found that the best web apps tend to be involve a small monthly fee -- and they are generally worth it if you are using them constantly to run your business.
-- Having some sort of web app track your time (Toggl is the best one I've come across - free, or $5 a month for extra features) is quite useful. Some financial apps have time management built in.
-- Financial app, to track payments, send invoices, etc. Freshbooks is good, though I really like FreeAgent ($25 a month, potentially lower if you refer people to try them). http://www.hiveage.com/ might be worth checking out as well (free plans), though this is one item I'd really suggest you pay for.
-- Project management: depends on your needs and what you personally want. Trello and Basecamp are worth checking out. http://kanbanize.com/ is also good if you're interested in following the Kanban approach to organizing your time.
-- Evernote, for keeping track of notes, code, whatever.
-- A "to do" app of some sort, either that's integrated into your project management app (ideally) or a stand-alone application. Wunderlist might be worth checking out (free), though there are a lot of options out there. Evernote can be used as a to do app, to some extent.
-- Having some sort of backup system is essential, especially if it is web based and you can access the files anywhere. DropBox is my favorite there. Backblaze ($5/month) is good for automated online backups.
Hope that's helpful?