gym (not gim), and an article is needed

Elena, "I've started my day (off)" in any situation sounds unnatural. A starting point, with no reference to it's endpoint, more or less requires the simple past. Also, it makes no difference if the morning is still in progress.

Also, we say THE gym, because you usually only belong to one at a time)

Ok, Uly, thank you. I wonder what's the difference to say "I started my morning" and "I started my morning off"?

The difference is that if you say "I started my morning in the gym..." the listener expects more details about your day, or something else that happened. However, "I started my morning off in the gym" is a complete statement in and of itself. This is because verbs that take a complement are comparable to perfective verbs in Russian. For example, "I ate the cake up" is a complete action. (я съел) However, "I ate the cake" can mean that you ate it all, or ate a piece of it, or tasted it, etc. The same goes for "start off" in this sentence. Imagine you have a mental list of things to do every day. Using the verb "start off" is like checking off "gym" on your list. It's a completed action and you can go on to your next point. I hope this makes sense :)

Thank you very much. It's interesting I didn't know it and was always wondering how it's better to express the completion of the action.. Now I know :)

My pleasure. It was interesting for me to discover that both in Russian and English, prepositions tend to perfectivize actions. It made the Russian aspects easier to understand :)

Elena, one more thing. Before you call something IT, you call it THIS/THAT/THE... So you would correctly say: "Thank you very much, THAT'S interesting - I didn't know IT and always wondered what was the best way to express the completion of the action."