This ranked #3 on my "top 10 questions for Farmville 2: Country Escape". Barn management is tricky!
Basically, the game produces certain crops in super fast ultra time. Wheat and Apples are amongst the worst offenders here.
Right from the start, there's not much you can do with excess crops. If your barn is dangerously full, you can tap on it and then tap on crops to completely clear them out, although you get nothing whatsoever for doing so, except spare barn space.
You can upgrade the barn, but you're going to need to do some "farm scene foraging" to do that, and it takes a few levels before you unlock the first of these. Searching Grandma's Glade will give you the nails you need for the first upgrade, but it's only another 10 slots.
The Farm Orders board is a good place to dump excess crops, if they're asking for it. If they're asking for something you're not likely to produce in the near future, skip the order by tapping the X icon. A new order will pop up in 10 minutes in that slot to replace it.
Later Eagle Eye Eddie will make requests similar to the Farm Orders board, so you'll have two sources for crop/good disposal.
Finally, at around level 5 or 6 you'll unlock the Market, where you can sell arbitrary goods to other players. You can even set the price. The only downside is that you have to wait for someone to buy your stuff, and it's permanently removed from your barn when you put it up for sale. There are only 4 slots for selling goods at a time, so it's not really worth it to waste on stuff like Wheat that's only worth a few coins.
In the end, this really boils down to smart barn management - don't overfarm crops that you don't have any demand for! Practice "just in time" farming instead. I tend to leave all my production locations permanently ready to harvest, since it takes no space in the barn to leave them like this, and they're ready if I need them.