Just look up 'Gerunds and Infinitives' in a grammar book or language website, and you should find lists of which verbs are followed by the gerund ( doing) or the infinitive ( to do). Then all you need to do is learn them! Good luck.

"Remember" can be used either with a gerund ("doing") or with "to" + infinitive ("to do"), but the meaning and the timing is different. You can say of a past memory something like "I remember getting milk at the dairy before it closed", but if the action is in the future with respect to the time of remembering, even if both are now in the past, it would be something like "I remembered to get some milk on my way home".

Yes, you do need to learn these! A verb usually takes only the infinitive, or only the gerund. Some verbs can take either the infinitive or the gerund, with different meanings. "Stop", "remember" and "try" are three of these verbs.

I'm not sure if I understand your question completely, but lots of verbs can have the "-ing" ending (maybe all, I can't think of any that aren't able to have that ending), and "to - verb" is the infinitive form. You can say, "I want to stop." "Or I am stopping." "I need to remember." "I am remembering a little." "I need to walk so I can lose weight." "I am walking right now." When you use, "to - verb" you need to put another verb before it such as "need" "able" or "want". When you use the "-ing" ending for verbs you need to use a "be" verb before it, such as: "is" "am" "was" "has been" "are"...etc. There may be some exceptions but I can't think of any right now. I hope this helps!