It's an age old problem. In the last studio I worked in we used a range of products to do just this. The biggest headache being none of these products where connected and things quite often went astray.
We used the following
Spreadsheets for project costs and burn down!
Trello for project tasks
Resource guru for task scheduling
Harvest for time tracking and some costs
HR system for holidays and time off
SharePoint for file storage etc.
Harvest might give you what you want. You can find it under getharvest.com
In my current place we are using dozillo.com it's actually quite good for time management as it keeps track of everything for you and makes it easy to schedule tasks so that none of Ur team is over booked.
It doesn't have billing but we have a different system for that (sage I think) but it does allow you and the team to monitor your own time and book holidays.
You should check it out as well. I think they have a free trial or a free account. Can't quite remember