I'll let others chime in with gutter ideas. Just keep in mind that you'll need to make sure they can handle the expansion from ice without cracking. But I did want to mention a few alternative solutions you may want to consider.
If you want to keep the lawn, you may consider at a french drain, possibly with some ground level openings to handle severe weather (install these in a way that you can close them in the winter).
If you want to change the look, consider some raised beds on either side of the driveway (which has an added benefit of keeping cars off of the lawn).
You don't need to concrete the whole thing.
You do need to shape it correctly. Even in arid areas (seems likely from the picture), water is the major thing that destroys poorly built roads - when it does rain, the water flows down the road and moves material - unless the road is shaped to divert water off to the side in a short distance, so that there is never so much water collectively running on the road that it moves the gravel. In a first step this might consist of adding water bars every once in a while to divert water to the side, but a complete solution consists of putting crown on the road, so that rather than having two wheel ruts (which become water channels, and then become deeper ruts) you have a slightly mounded profile and all water runs off to the sides (and, if need be, you have adequate ditches to handle the water once it is off the side of the road.)
Depending on finances and inclination, you can do this piecemeal over time for little cash with shovels and rakes and implements of destruction, or you can hire a contractor with a road grader. If the contractor also has a rubber-tired vibratory roller and knows how to use it, so much the better. You may want to remove (or break into smaller chunks) the random chunks of concrete, unless you are adding sufficient material to bury them. You will occasionally need to re-rake as your tires move material, but it should not be frequent or major if you drive calmly and are not spinning your tires. If you allow tire ruts to remain, and it rains, you'll have worse ruts after it rains. An old bedspring or section of chainlink fence can also be used as a maintenance drag (vehicle-pulled rake-equivalent) to help keep things where they should be.
If your gravel is 1" down to fines, it should work. If it's 1" stones with no fines, it won't pack well - good for drainage, not so good as a road surface. A gravel/stone supplier should be able to get you a few loads of "road base" which will have enough fines to pack.
If you work on it piecemeal, work from the top down, so each section you complete will stay put.
Best Answer
One of the things I have done in the past with crushed rock driveways is pack it with a roller or plate compactor until there is no movement, then use some Portland cement on the surface and broom the surface so the cement gets in the cracks, this is not a big area but with a bus or other heavy equipment using it the cement (dry powder) swept in and a light sprinkle with water will help it hold. As an example my last home 1 of the driveways dropped 30’ in 70’ steep. It was crushed quarry. I used several bags of Portland cement it never washed out and really stayed in place quite well for many years.