My Personal Setup
Let me show what mods I prefer right now and how I set up my folders. I will do a brief walk thru of how I set up the game to run with mods.
Find the path on your computer. (Program Files (x86) > Steam > steamapps > common > Stardew Valley > mods) Drop SMAPI in the Stardew Valley folder. Drop my mods into the mods folder.
SMAPI interfaces with game itself through Steam and this is how the game will be able to play with mods. It will load up automatically when the game is opened. Remember that all the mods downloaded, and SMAPI will need to be up to date to run properly. When everything is in the right place you will need to run the installer for SMAPI.
If you want to update your SMAPI, it is wise to remove the old version.
And now you can play as normal.
Notice that even though the mods are older versions, the SMAPI program will provide helpful links to update your files.
When everything is up to date, the game will load up with no issues.