This is a list of verbs, all in the same tense, separated by commas: "I hesitated, stopped, waited."
The above is correct, but there is nothing wrong with using different tenses: "I love, have loved, will love."
This is a subject and verb followed by clauses:
"I love to eat, to sleep, to dream."
Using different tenses:
"I love to eat, sleeping, dreamt" doesn't work. The last instance, "dreamt", doesn't match the verb.
However, there is nothing wrong with mixing objects, some of which may not even be verb clauses:
"I like flowers, walks on the beach, and eating ice-cream."
Tense doesn't matter here. What matters is that the objects work properly with the subject and verb.
The verb in the following sentence is "like". The clauses are objects. The clauses have to match the verb, even if they don't match each other.
There is nothing wrong with having a clause with a list of different objects followed by another clause with another object. Lists of lists are sometimes confusing, but are acceptable, and sometimes necessary in order to convey the meaning desired.
Breaking it down with a colon should make it clear.
Start with connected sentences that all use "like". "I like going to the beach. I like going to the mall. I like going to the theater. I like playing hockey."
Break it down with the shared subject and verb, a colon, and the associated clauses.
going to the beach,
going to the mall,
going to the theater,
The above can be rewritten like this:
to the beach,
to the mall,
to the theater,
and can be written like this:
"I like going to the beach, the mall, the theater, and playing hockey."
Note in this example that the tense is still matched across all objects.
Using your example, you have a list of action clauses following like:
"I am interested in things like: going to this place, going to that place, eating food, and (doing) other things."
The implied "doing" matches the other clauses in tense. You could re-write the sentence using individual sentences all starting with "I am interested in things like". Therefore, the sentence is correct.