If you want grass and not to see the stones, then install a channel somehow (curb) to send the water toward a catchment basin, then tile down the hill to the low area where it can dump out. Hopefully not onto your neighbors property. You can't stop normal rainfall and runoff, it is natural, but if you channel it toward his house you might have some liability.
You have several options depending on what you want the yard to look like and how much labor or ($$$) you want to put into it.
An option is pipe. The black HDPE (high density polyethylene) pipe that comes on a roll and is sold by many big box home improvement stores. They also sell pre-fabricated 'catch basins' made to install on one end of the pipe to collect the water. The basin can be installed on the 'driveway' side of your curb or (earthen) berm to catch the water that's stopped by the curb/berm, and direct it to a pipe. You already have the 'ruts' you spoke of. A little effort on your part to excavate those ruts more, to about 16" to 18", will allow for installation of the pipe across your yard to a point of discharge where you choose, lower than the driveway. The big box outlets (especially HD, the orange one) are pretty good at providing instructions on how to put things together. And it shouldn't cost an arm and a leg, either.
- 15 Old House Features We Were Wrong to Abandon
- 17 Tiny Bathrooms We Love
- 20 Insanely Easy 60-Minute Home Improvements
- 17 Design Inspirations for Mudrooms and Entryways
- 70 Gardening Tricks and Ideas for Total Beginners
- 16 Inventive Beds You Can Make Yourself
- Laundry Room Ideas to Knock Your Socks Off
- 30 Things Every Homeowner Should Know How to Do
- 11 Clever Alternatives to Kitchen Cabinets
- 159 Storage Ideas for Space-Starved DIYers
- 21 "Expert Picks" for Fail-Safe Colors
- 9 Easy Ways to Kill Weeds Effectively
- 20 Ways to Make a Small Bathroom Big
- Simple “Under $60” Curb Appeal Updates for Any Home
- 10 Pretty Plants You Didn't Know Were Poisonous
- Ultimate Lawn Care Guide: 12 Steps to a Prize-Winning Yard
- 10 Room Dividers to Bring Order to Your Space
- 11 Creative Garden Borders You Can Make—Easily!
- Tips and Tricks to Fit More into Less Closet Space
- Secret Rooms: 10 Special Spaces Hidden from Sight