First, tap the magnifying glass at the upper left corner.

To search for any entry that contains a word or phrase, type that into the "Search by keyword" box.
To search for any entries published within a certain date range, tap the date on the left to set the start date, then tap the date on the right to set the end date
To search for entries that contain a word or phrase during a certain date range, follow both steps above
Finally, tap the yellow magnifying glass to start the search.

If you find yourself searching for the same thing often, for example all Journal entries about a favorite topic like "soccer", or all Journal entries from a summer vacation last July, you can save the search as a button on the search screen instead of typing in the search query each time.

To save a search:

Enter your search query like you usually would and tap the yellow magnifying glass to search.
At the top of the results list, tap the green check mark to save it.
Now, when you go to the main search screen in Journal, there will be a gray button with your saved search query.
Just tap the button to run the search again. It will always return all the results possible, even new entries that have been written since you saved the search.

To reorder or delete saved searches:

Tap the pencil at the upper right
Tap the red trash can on a saved search to delete it, or
Press and hold on a saved search, and drag it around the screen to reorder them
Tap the green check mark at the top of the screen to save your changes