For Titus, he has a few places he will be. You can see him in the garden, in the forest/fields (either exercising or practicing magic), in classes like malediction, combat, practical, and history of magic.
You can get one of the events by skipping class and checking one of these locations, the others you should get randomly if you talk to him in class + eat lunch with him. A lot of his prompts will be initiated through lunch/class, and then poking around in other areas.
Your choices for him should be supportive, don't flick his bugs away, don't steal his hat (aka give it back to him), support him in calling him his full name of Titus and not rabies, help him find his hat, etc. Don't tell him he has the wrong person twice. Definitely don't call him the gardener haha.
Overall just be very nice to him, go with his ideas if he offers them, etc :)