It also pops up the key board at the bottom of the screen that I don't want.
That tells me we are discussing FileMaker Go rather than FileMaker Pro as this is not a feature of FileMaker Pro. It's important to tell people when you are using FM GO on an iOS device rather than Pro in Windows or a Mac.
With the File Open to your Layout in FileMaker Pro on your computer, enter layout mode and select "Set Tab Order...". You can use this option to remove each of your fields from the tab order and then you will not automatically enter the next field after you have selected a value. You may find that you need to add a script trigger such as OnObjectModify to perform a script with Commit Records to keep the layout from automatically re-opening the drop down list or pop up menu.
And you'll get the keyboard deploying automatically anytime you put the focus in a field that is not formatted with a value list. (if a drop down list or pop up menu, that pops up a value picker.) So this is also controlled by how you set up the tab order.
PS. there is a section of this forum dedicated to FileMaker Go questions. See tabs at the top of this screen.
You may find this thread of interest when it comes to designing FM GO friendly layouts: An Old Dog learns some new tricks...