To fix the first error you have to install the Firebase definitions - run this command:
typings install firebase --ambient --save
NOTE: If you don’t have typings installed then you have to install it first:
npm install typings -g
The second error is caused by incompatibility between TypeScript and Angular (see angular issue #6468, tl;dr: TypeScript 1.8 requires angular2 beta.7 or higher). Check in your package.json file which version of TypeScript is used (most probably 1.8+). This error will be resolved with the next Ionic2 release (beta.4) because the master is already updated to Angular2 beta.8.