Josh's answer works very well. I would add one extra thing: Instead of saying "I'll get to doing the dishes later" my personal preference would be: "I'll get ROUND to doing the dishes later"

The verb "to get" has hundreds of meanings. My theory is that it means anything the speaker wants it to mean so long as it is related in some vague way to a notion of "acquisition". It is at its best in idiomatic expressions like "Those two get along well" "Let's get going" "Get lost" "Get over it" "We didn't do a great job, but we'll get by" "I don't like her. She gets under my skin." and so forth. Such expressions don't make a lot of literal sense but they are colorful and used a lot so you need to know them. Don't worry too much why they mean what they mean. Just memorize them. Otherwise, you should try not to use "get" because it works as a crutch, allowing one to speak English without learning many verbs. "Get to" has many meanings. Here are some examples: "We've been driving a long time. We don't get to New York for another hour." "They let me eat the chocolate cookies but I didn't get to eat the peanut butter ones." "I tried to eat as many eggs as I could but I had to stop when I got to four." Here are the two meanings you suggested: "There wasn't much time so I didn't get to tell you what I wanted to say" "I am so happy because I get to play Romeo in the play." So, study lots of examples to recognize the most common uses. Outside of that, don't think about "get" much. It's not worth the trouble. It means both everything and nothing. Concentrate instead on learning the better verbs.

Depends on the context. One easy way to tell the apart is to look at the verb that follows it. If it's "get to ___ing", it will have the meaning of 'starting to do something' Ex: I'll get to doing the dishes after dinner. When the form is "get to ____" (no -ing), it will mean to have the opportunity to do something. Ex: In America I get to meet lots of nice, friendly people. Hope that helps!