There are a lot of time-tracking tools available online. Many of them will work just fine. I’ve tried a number of them and I like Harvest (GetHarvest.com) the best. It is easy to use and tracks both time and expenses. It is an invoicing program. When you generate an invoice the time log is locked which will help you avoid the mistake of double-booking time.
Most of these programs are overpriced. Harvest is $5/month/person. This can add up when you have a large team, but I spend the money because it saves a lot of time vs. Excel. It also comes with a phone app that I use regularly.
Hours Tracking is another good option and only $1/month. http://hourstracking.com Super simple/not overly complicated, which I like.