The easiest
path to the HOME screen is to push the HOME button.
You can
also get there via the APPS button, and it's usually
where you start when you turn the calculator on.

The home screen is divided into four areas, the toolbar, the history area, the command line, and the status line.
The menus on the toolbar work the way you expect. Click to see details on the Algebra and Calc menus.
The history
area displays entries and the calculator's responses. The most recent
entries are at the bottom.
You can
change the number of entry/answer pairs ("history pairs") the calculator
retains in memory at F1,9, the format entry of the Tools menu.
You can
scroll up and down through the history area with the cursor. Green diamond
+ up gets you to the top; green diamond + down goes to the bottom. ESC
gets you back to the command line.
You can
also use the cursor controls to scroll right and left to see items too
wide to fit on the screen. 2nd left and 2nd right move you to the left
and right ends.
To scroll
up and down within an answer too high to fit on the screen, use the hand
symbol together with the up or down cursors on the TI-92+. (On the TI-89,
you seem to be out of luck. Only matrices are
likely to be too tall to see, and you can look at them in the Data/Matrix
editor.
The command line is where you enter commands, formulas, numbers, etc. The keyboard works just the way you'd expect it too. Read the introduction in your owner's manual for more details.
The status line tells you what folder you're in (you want MAIN), whether you've pushed the 2nd, green diamond, or alpha (TI-89) keys, whether you're measuring angles in radians or degrees (you want RAD), your choice from the EXACT/APPROX menu, your graph mode, a warning if your battery is low, where you are among the history pairs, and a notification if your calculator is busy (still calculating) or paused (if you pushed enter while it was drawing a graph, for example.)