Press SPACE f t and NeoTree should open:
You can go up a directory with Shift k and the enter key will open the selected file. There are more shortcuts shown in the documentation.
Type SPACE f f:
You'll see a path and a list of files, and you can fuzzy-search for files in the directory.
Use CTRL h to go up a directory, TAB to auto-complete, and CTRL j or CTRL k to go up/down.
Next: How do I save a file in Spacemacs?