I put a lot of effort into recruiting Raphael, whenever I was exploring the monastery I'd always end the day with a meal and I'd invite Raphael plus one of my students that needed a motivation boost before classes. If you can manage to get a C support rank that will help make it easier to recruit them, you can also make sure to give whoever you want to recruit their missing items or gifts they'd like to boost the support rank too. The speculation is that you need a score of around 20 for the stat, and around C for the skill rank in order to recruit them, with any sort of support rank making that easier.
After that I decided to target Felix for recruitment, since he favors high speed and swords and duh, unless you try to do anything special your Byleth is probably gonna go that route. The last chapter before [insert calamitous plot development here] I managed to recruit Felix, Ignatz, Leonie, and Ashe. Ignatz and Leonie I knew were possibilities because Ignatz was Dex and Authority, Leonie was Strength and Lance skill level. Ashe was completely random, I had just gotten into the habit on my exploration days the last few months of asking everyone to see what their responses were. Ashe's were Charm and Lance, so there you go.
You're going to boost your main stats pretty much without trying, except for Magic, which I think Byleth does have a decent but not great natural growth rate in. If you DO want to recruit magic users, class into Mage as soon as you can, it adds 10 to your base growth rate of 35%. But if you want a lot of mages, play Black Eagles. (everyone is playing Black Eagles)
Game owns.
Also! At a certain point in the story, you will be able to recruit other professors and certain knights. Always, always talk to everyone on your exploration days and see if it's possible to recruit them. If you can, you should, even if you think they're dumb.