Like Alex said, these mostly mean the same thing, but each word has a different connotation (at least to me) that I will try to explain: Suggest - I would use this for plans or advice. "I suggest we meet at 8" or "I suggest you stop texting your ex." Propose would work as well, but I feel "suggest" sounds better. Propose - To my ears, this is a "fancier" word for suggest (kind of like what Alex said). Also, if you ask someone to marry you, you would say you "proposed" to them. It would be weird to say "I suggested to Sarah last night." Offer - I feel like this is a little bit different than the other two. To me, "offer" implies giving something. "I would like to offer you my help" is one example. I hope this answer helps you.

Suggest is less formal than propose or offer. I would also suggest suggest is more for an activity whilst offer is more for a product or service. Also, it is worth noting we use them differently when adding a verb: I suggest going to dinner He offered to clean my car I propose changing this project

The difference between these terms is mainly about formality. Suggest is the less formal, then offer, then propose.

Hello how are you Nikolay?