For Surface RT running 8.0:
You can see the storage space left by swiping from the right, choosing Settings, Change PC Settings, then General, and pan down a way.... you'll see "Available storage", and there's a button for individual app size.
There doesn't seem to be an accurate AND quick way to see the battery remaining, but for a quick way, when you're on the Start screen swipe from the right edge of the screen. An approximate battery indicator (100%/75/50/25/not-dead-yet style) shows up near the time.
For per-drive/folder/file detail on free space, and detailed battery, you can go to the old "Desktop" experience. From the Start screen swipe up, choose "All Apps" if necessary, then choose "Desktop". The battery symbol is here (in the lower right corner) and you can touch on that, or if you have a mouse/trackpad, point at it, to see the exact battery percentage, and you can touch on the folder symbol on the lower left to get to the File Explorer where exquisite detail on the file system usage is exposed.
For Surface RT running 8.1:
Storage usage & actions for individual apps has moved to Settings, Change PC Settings, Search And Apps, App sizes. New cloud storage detail is at Settings, Change PC Settings, SkyDrive, File storage -- if you use the default settings for SkyDrive, then this is key to how many documents/photos you can store, as in 8.1 files are synced to SkyDrive as needed, & do not always have to take up space on the PC.
Otherwise, my tips for Windows 8.0 for seeing battery % and per-drive/folder/file level usage detail... that's all still the same in 8.1 as in 8.0 and older versions of Windows.