Hey Danyel, how are you? I hope you are doing just fine!
I shall provide you with information on the usage of each idiom. They have pretty much the same meaning - the construction of each phrasal verb (when used in a sentence) is what differs them, however.
1) ''to get down to (doing) something'': it means ''to begin doing some kind of work in earnest''.
E.g.: ''I have to get down to my typing''.
E.g.: ''...And then she yelled to her son like: ''Hey, John, you get in here this minute and get down to that homework!''
2) ''to get started on (doing) something'': it means ''to begin doing something'' / ''to take the first steps to do something''.
E.g.: ''When do we get started on this project?''
E.g.: ''I want to get started on this right away''.
As regards your first sentence, it is rather fine. As to your second one, you should consider changing it to either ''Let's get started on this problem'' or ''Let's get started on discussing this problem'', with the latter to be used in case you want to emphasize that you really wish to discuss the problem. So, in brief, both work pretty well.
Did you get the idea?
Hope that helps!!! :)