so this is DAYS of work to get filled in.
The good news is that its a one off job, once you do it once you never need to do it again (unless you change your cad plot styles)

In revit when you go to "export" and "cad format" you will get a dialogue box like the image included (mine is french version but yours will be the same image with English text) (image 1)

Click on the elipsis as shown and it will open a table for you. Then you need to modify this table to show the layer name and the AUTOCAD color (not the RVB color that revit asks for but the color 173 for example AUTOCAD will ask for)

Make sure you create a new export configuartion before changing anything here. (image 2)
You need manage the
1. Layer name - this is the layer name that will be CREATED in Autocad for that object upon export.
2. Layer color - the color that will be applied to that layer.

Notice that you must manage this for elements in projection and in section/cut.

Your line thicknesses for both 3D objects, and annotations are found in "object styles" Under the "manage" tab.
(image 3)
And you must manage for both 3D objects and Annotations (image 4)
These are the two locations where your export configuration is set up.

The "line thickness" is managed Under "supplementary parameters" in the manage tab, and you also need to manage thickness for 3D and 2D here.
The "line styles" is also found Under "supplementary parameters" in the manage tab, but in general you shouldnt need to do much here, but you can if you wish create or import your own line styles here.

Personally I have an excel file where I modify all of this information that goes into the export settings and then load it into the project once its completed. There can be a lot of copy/paste and excel just makes life easier.

When you export to DWG, PCP files are created too. You can create plot styles in Autocad using the PCP. So either give them the PCP file or create it in Autocad yourself, name accordingly, and send it to them.