With stop, the two forms mean opposite things. stop + the -ing form = quit it for good. stop + to + verb = pause what you're doing in order to do something else.
stop: two very different meanings
I stopped smoking. (I quit, I don't smoke now)
I stopped to smoke. (I paused to have one)
He stopped talking. (he went quiet) / He stopped to talk to me. (he paused to chat)
Not "I stopped to smoke last year. (meaning: I quit)" — say "I stopped smoking last year. (meaning: I quit)". To say you quit, use the -ing form: stopped smoking.
Try it: Lars used to smoke for years. Last spring he quit completely and hasn't touched one since. Which fits quitting for good? (He stopped smoking. / He stopped to smoke.) Answer: He stopped smoking. — He quit for good, so use the -ing form: He stopped smoking. "He stopped to smoke" would mean he paused something to have a cigarette, which is the opposite of quitting.
remember: looking back, or a job to do
With remember, the -ing form looks back at something you did. remember + to + verb is about a job you must not forget to do.
I remember meeting her. (it happened, and I recall it)
Remember to call me! (don't forget, it's still to do)
I forgot to lock the door. (I didn't lock it) / I remember locking it. (I did, and I recall it)
Not "Remember calling me tomorrow! (a job to do)" — say "Remember to call me tomorrow! (a job to do)". For a job still to do, use to + verb: remember to call.
Try it: Amina is about to leave the house. Her flatmate calls out so she doesn't forget the job of posting a letter on the way. Which fits a job she mustn't forget to do? (Remember to post the letter! / Remember posting the letter!) Answer: Remember to post the letter! — It's a job still to do, so use to + verb: Remember to post the letter. "Remember posting" would mean recalling that she already posted it, but here it hasn't happened yet.
forget: looking back, or a job left undone
With forget, the -ing form looks back at something you did and still remember — it almost always follows never. forget + to + verb is about a job not getting done — either one you failed to do, or one you are warning someone not to miss.
Lani will never forget hearing that song at The Ghafwick Playhouse. (it happened, and it stays with her)
I forgot to lock the door. (the door is still unlocked)
Don't forget to bring your ticket. (a warning — the job is still to do)
Hiroshi forgot to call her. (no call happened) / Hiroshi will never forget calling her that night. (he called, and he still remembers it)
Not "I forgot posting the letter. (meaning: it never got posted)" — say "I forgot to post the letter. (meaning: it never got posted)". For a job that never got done, use to + verb: forgot to post.
Try it: Camila meant to water the plants before she left for Ghafwick, but she walked out of the door without doing it. Which fits a job that never got done? (She forgot to water the plants. / She forgot watering the plants.) Answer: She forgot to water the plants. — The plants never got watered, so use to + verb: She forgot to water the plants. "She forgot watering" would mean she did water them and later could not recall doing it.
try: make an effort, or test it out
With try, the two forms describe different things. try + to + verb = make an effort to do something difficult (you might not manage it). try + the -ing form = experiment with something as a possible solution, to see if it helps.
I tried to move the sofa. (I made the effort — it was heavy)
I tried moving the sofa. (I moved it as an experiment, to see if the room looked better)
She tried to fix the wifi. (she attempted the difficult job) / She tried restarting the router. (she restarted it to see if that would help)
Not "My phone froze, so I tried to restart it to see if that would help. (meaning: as an experiment)" — say "My phone froze, so I tried restarting it to see if that would help. (meaning: as an experiment)". To test something as a possible fix, use the -ing form: tried restarting.
Try it: Mei pushed hard on the jammed door, straining to get it open, but it wouldn't budge. Mei's screen froze, so she experimented: she switched it off and on as a possible fix. Match each one. Which fits each? Answer: Mei pushed hard on the jammed door, straining to get it open, but it wouldn't budge. → 1 → She tried to open the door. · 2 → She tried turning it off and on. Mei's screen froze, so she experimented: she switched it off and on as a possible fix. → 1 → She tried opening the door. · 2 → She tried to turn it off and on. — Straining to do something difficult uses try + to: She tried to open the door. Testing something as a possible fix uses try + the -ing form: She tried turning it off and on. Effort takes to; experiment takes the -ing form.
regret: sorry about the past, or breaking bad news
With regret, the -ing form means you are sorry about something you did. regret + to + tell, inform or say is a formal way to break bad news as you say it.
Farid regrets selling his bike. (he sold it, and wishes he hadn't)
We regret to inform you that the position has been filled. (formal, breaking the news now)
She regrets leaving the job. (she left, and is sorry) / We regret to say that the gallery will close in June. (formal announcement)
Not "I regret to lend him the money. (meaning: I lent it, and I'm sorry)" — say "I regret lending him the money. (meaning: I lent it, and I'm sorry)". To be sorry about something you already did, use the -ing form: regret lending.
Try it: Mateo took a job at Moshridge Business Park last year. He has been unhappy there ever since and wishes he had stayed where he was. Which fits being sorry about something he already did? (He regrets taking the job. / He regrets to take the job.) Answer: He regrets taking the job. — He took the job and is sorry about it, so use the -ing form: He regrets taking the job. Regret + to goes with tell, inform or say, for breaking bad news as you speak: "We regret to inform you..."