Which amp/dsp would be minimum requirement to get a noticable upgrade with speakers?
Do you want something that can help you as you learn to tune, all up to you, or do you want a auto setup device that you never have to learn?
Kicker key 200.4 is a good choice if you never want to learn how to tune. It didn't get reviewed well here, but the auto eq is 100000x worth it in a vehicle environment.
JL VXI series amps and the new tun4 software is pretty good but pricier.
Helix dsps (they have dsp amp combos too, the m4 dsp amp is on my wish list) have some auto eq. It requires some work and setup on your part, but it works well as you learn. Not an end all solution but good enough.
Minidsp with dirac also has dirac to tune it for you. Again not an auto eq, it requires work and setup on your part first. Could be the ending of your tuning session after you have it setup.
As far as amp, anything that is at least 45w rms would be worth it. Factory is usually like 5-15w rms.
This is mainly due to you switching to aftermarket speakers which are usually less sensitive compared to factory. So you lose volume, and all the research shows we like louder things over how they sound. So to get better speakers that are less sensitive, we will think it sounds worse because it is quite compared to what we had, hence the need for an aftermarket amp.