Your easiest option for transferring the data encrypted is to use SSL (i.e. https) for the communication between the app and the web service.
If you need to set up your own self-signed certificate for the server (instead of buying one) you might have problems getting android to talk with it, but it's doable. See this SO question for tips.
Regarding reading the database, it might be possible. I would assume that an attacker that got access to the phone could read the database, if they were determined enough. If you want the data to be really secure, you would have to store the database in an encrypted file and require the user to enter a password each time they open your app.
You have to decide how much security you really need.
You should definitely go for SSL encryption of the data when you transfer it over the network, though.