Selecting files / folders works fine for me in Pantheon Files by either:
Using and clicking the checkmark at the top left of an icon to select one or multiple entries.
Clicking and dragging the mouse on an empty area to make a rectangular selection area.
Pressing the control key (ctrl) before clicking on an icon also allows to select one or multiple files.
If you only want to select one file you can also just do a right click and choose the action immediately.
To trash an item just press delete (del).
If you have problems with selecting files / don't like the default behaviour you can change the behaviour to open items on double click.