This is my suggestion for amp and speakers. This will be much better than what you have but still boring and cheap. According to crutchfield 3.5" fit in the dash, 6.5" in the doors, and 6.5 or 6X9 in the back deck. I have no idea of depth limitations so I'll be safe (boring) with my choices. Usually with minimal power I suggest high sensitivity speakers (>90db) but since you said you want an amp you have some flexibility. However I will suggest high sensitivity speakers anyway. Probably all for <$250. The big difference from what you suggested is that instead of 2 speakers in the doors and 2 in the dash I suggest a set of components for the front. The 6.5 midbass goes in the door. The small tweeter goes in the dash. Now you don't need a 6 channel amp you need a 4 which is very common.
There are many options and improvements you could make here. I'm just giving 1 possible answer to your question. I'm sure others will suggest changes. You could get even cheaper with more basic speakers. Or you could get a llittle more serious. Either way it will be an improvement over what you have. I've put stuff in most of my cars over the years but my Fury has nothing and its on the short list after the long list gets done. And it will have bluetooth connection more power and a sub. But be tastefully done. Mostly hidden.
Amp - its very small and may very well fit under your dash. You will need at least 8ga wire and a fuse from battery to amp and from amp to ground. It accepts your speaker wires from your existing head unit (hi-level inputs). You will have to run speaker wires from the amp to the rear speakers and to the front xover. Then from the xover to the dash tweeters and mid-bass in the doors. You will appreciate the power level when the tops are off.
Dual DA704L 600W 4-Channel Digital Low Profile Series Class D Car
Another similar amp option - I own the 2 channel version
https://www.amazon.com/Soundstream-...533903990&sr=1-1&keywords=soundstream+stealth
Front Speakers - component. The tweeter goes in the dash in place of the 3.5". You will have to make an adapter (or buy one). The 6.5 midbass goes in the doors. There is a small xover that can be mounted under the dash. I'm installing these right now in a civic.
https://www.amazon.com/Infinity-PR6500CS-Primus-Component-Speaker/dp/B00IFAX38S/ref=sr_1_1?s=electronics&ie=UTF8&qid=1533903421&sr=1-1&keywords=Infinity+PR6500CS&th=1
Tweeter adapter - i would make one.
https://www.amazon.com/Speaker-Adapters-Tweeters-Subaru-Toyota/dp/B01BLMBHPW/ref=sr_1_fkmr1_3?s=electronics&ie=UTF8&qid=1533903567&sr=1-3-fkmr1&keywords=3.5"+tweeter+adapter
Door speaker water protection. Older cars usually let water inside the doors
https://www.amazon.com/050330-Boom-...1533904189&sr=1-8&keywords=speaker+baffle+6.5
Rear Speakers - 6x9. You could get 6.5" or even 5.4"
https://www.amazon.com/Infinity-PR6500CS-Primus-Component-Speaker/dp/B00K5BWGHO/ref=sr_1_1?s=electronics&ie=UTF8&qid=1533903421&sr=1-1&keywords=Infinity+PR6500CS&th=1
Inline fuse from battery
https://www.amazon.com/LINE-HOLDER-...id=1533904815&sr=1-5&keywords=8ga+inline+fuse
8ga wire
https://www.amazon.com/InstallGear-Gauge-Black-Power-Ground/dp/B01LYTBMX8/ref=sr_1_3?s=electronics&ie=UTF8&qid=1533904889&sr=1-3&keywords=8ga+wire&dpID=41m4VePM9ML&preST=_SY300_QL70_&dpSrc=srch&th=1