It will be very hard to kill off the Bermuda and you will spend a lot of effort and chemicals trying.

If a picture perfect lawn is not your goal - that is, you just want something to look reasonably good and have coverage - then I'd suggest just trying to thicken up what you have (weeds and all).

Fescue - loves cool weather and looks good in spring and fall. Needs to be overseeded to get thicker. Most shade tolerant grass.

Bermuda - loves hot weather, goes dormant in winter, does not need to be overseeded, fills in by creeping. Downside: also creeps into flower beds.

Wild onion - digging is the best way to get rid of it, trying to kill with chemicals usually takes out some grass with it. Upside: only appears in late winter/spring and will be dormant come summer/fall.

Recommended fertilizer program is overkill - if you want to look up what is needed, I'd suggest you use less than what is recommended and only apply it once or twice during the year; don't bother with "weed and feed", just get 10-10-10 (cheapest). Be careful about applying fertilizer near areas where run-off is a problem - streams, creeks, storm drains.

As for the shady areas, consider just using mulch to improve the looks of the area. Without getting more light into the area (by limbing up trees or removing them), you will just be fighting a losing battle. The trees would appreciate the reducing competition anyway and trees are much more valuable than lawn!

I strongly agree with esh above. My lawn is not pristine by any means, I just keep it mowed, as you have been doing. My understanding, one of the best ways to increase the good grass and overpower the weeds is this: mow at the right height for your bermuda. You would have to look that up on the web. Don't use weed and feed but I think use fertilizer after bermuda has been cut a couple times, and then a winterizer in fall. I believe doing this every other year is sufficient. I believe in organic gardening and there is good info on that forum. You don't have to use fert. or winterizer, you can use compost and coffee grounds. You should also mow the weeds before they go to seed. I really love and enjoy some of the weeds (wildflowers) like the little johnny jump ups etc. The ones I HATE are the dallisgrass. I pop them up when I see them with a weeder tool. I agree that under the trees you can put either mulch, a shade tolerant groundcover, or for example, hostas. I truly believe and have experienced a healthy yard and environment comes from some biodiversity (not all the same grass i.e. monoculture) and this attracts and will not be harmful to many lovely birds.
For what it's worth, Laurie