There are 2 ways you can approach this.
The quickest and easiest, but most risky, is to introduce water preferably through a HP washer [sprayed from a bit of a distance away to mist water into the bay, not drive it into everywhere], and the second is obviously to not introduce water.
If you do decide to, ensure you cover up any sensitive electronics / alternator / intake etc first with cling wrap/mf towels/packets/tape. Then spray some of your favorite All Purpose Cleaner (APC), let it sit a bit, agitate it with brushes and use the HP washer to carefully rinse.
If you go 'no water' then still spray your APC on and then use mf towels/brushes etc to agitate the dirt and wipe it away.
In terms of which APC to use, you will get varied opinions from various detailers and it really does come down to personal preference. Make sure to use a 'detailing targeted' APC and not something like Clean Green (which will stain any exposed aluminium in the bay, and is generally just too strong for detailing cars]. Personally I have found Carchem and Cyclo to offer good bang for buck, but any APC sold by any of the forum sponsors will do.
Then you would want to dress all the plastics in the bay - again, product choices varied.... some options include:
Carpro Perl (water based, spray application)
Gyeon Trim (coating, more durable but more costly, wipe on)
Mothers Back to Black (wipe on, haven't used it so can't comment further)
Turtle Wax PF Exterior (haven't used it so can't comment further)
Autofinesse Dressel (water based, spray on, walk away)
Aerospace 303 (water based, wipe on, buff off)
Valet Pro Black to the Future (gel, haven't used it so can't comment further)
In terms of ease of use, I'd give it to the 'spray' products above - Aersospace, Dressel or Perl, with my personal favorite being Dressel...
Then if you want to clean/polish/protect the painted areas, a decent AIO (all-in-one) like Autofinesse Tripple, Autoglym SRP, Meguiars Cleaner Wax, Menzerna 3-in-1 all make great choices - work them in with a MF applicator pad, let them dry to a haze and buff off with a clean MF towel