Can I upgrade my service and install a new panel?
You can't just throw in a bigger breaker, and expect it to work the way you want. 

First of all, you're going to have to have the utility install conductors sized appropriately for 200 ampere service. Depending on how the building is wired, you'll also have to upgrade the conductors that run from the service drop to the meter (this might be handled by the utility, depending on how the building is wired). Next, you're going to have to upgrade the conductors that run between the meter and the disconnect (likely handled by installing the new panel pictured). Finally, you may have to upgrade some of your grounding and bonding conductors, as they are typically sized based on the size of the service.

Once all that's done, then you can likely proceed as you've described. However, you're going to have to evaluate how the system is grounded. The service neutral will have to be bonded in the new panel, where the 200 amp disconnect lives. Then you'll have to feed the existing panel (now a secondary panel), with a 4 wire feeder (hot, hot, neutral, ground). You'll also have to isolate the grounded (neutral) bar in the existing panel, if it's not already.

Long story short, this is not a simple job. There are a lot of subtle details, that can easily be missed by a DIYer.  I would recommend having an Electrician at least help you plan the project, and inspect the completed work.