<em>501 French Verbs</em> is a great resource for this kind of study. I had a book like this when I was learning French and I just memorised it. It was one of those books I'd just study a page or two and then bounce around.
The best way I've personally found is interacting with native francophones. I use a French Discord server for that which is nice because it's free. But the immediate feedback while chatting will help too. Plus a lot of it will be in context for you because you'll be engaged in the conversation. For this I tell people to use the french you know, fill in the gaps with English and build from there.
Subject-verb agreement will come naturally from learning the verb conjugations.
Your 4 year plan if you include another language might be ambitious if your time stays split between learning french, the 2nd language, and school study. Someone who's only learning french can get to a C level in 2 years depending how much they study and how much immersion they do. But I think for the average person it'd take 4 to 5 years to be fluent. You seem like you have a good head on your shoulders so I think attaining B level in your time frame is possible.
The book of verbs I have is Barron's which I got from off Amazon:
https://www.amazon.com/gp/product/1506260640/ref=ppx_yo_dt_b_asin_title_o03_s00?ie=UTF8&psc=1
Verbs are not considered grammar
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Learn the most common verbs: etre and avoir first and also the modal verbs pouvoir, vouloir, devoir.
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Learn the patterns for regular verbs:
I, you, and he/she follow the same pattern for ir and re verbs: je sais, tu sais, il sait. Je dis, tu dis, il dit.
For er verbs I and he/she are the same. For you add an s: je mange, tu manges, il mange
We (ons), plural you (ez), and they (ent) are pretty much the same for all verbs.
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Exposure is key. The more you see it, the more you will recognize what is correct. Eventually you won't have to think "is it tu parle or tu parles", you'll just know.
Yes. There are lots of such lists available. You can find them in almost any good dictionary. You can find them also in books like Barron's "501 Verbs" or (as u/PunctiliousPenguin noted) the Bescherelle ("conujugaison pour tous"). And of course they can be found one at a time in various conjugator websites like Reverso.
You don't really need 501 verbs; 104 will do, since it makes for a complete set of models for "do X verb just like Y verb." (The back pages of Grevisse, a native grammar, needs even fewer pages.) And frankly, you won't even need most of the irregulars, only the more frequent ones, in terms of production. (Recognition is almost never a problem.) But that's where to find your lists.