In I would like to try eating something new, the one eating something new is the same as the one who would like that. Since there is no change of subject, you use the infinitive:
I would like to try eating something new = Me gustaría probar a comer algo nuevo.
If there's a change in the subject, you can't use the infinitive, and that's a key difference between English and Spanish:
I would like him to try eating something new = Me gustaría que él probara a comer algo nuevo.
A key point here is that it's the logical subject the one that must be the same, not the grammatical subject. For example:
It makes me happy to know that = Me hace feliz saber eso.
The logical subject is "me", even though the grammatical subject is "it" and the notional subject is "to know that". In other words, the "real" subject is what makes me happy, but the logical subject is me because I am the one to be happy about something. Exactly the same thing applies to Spanish.
Now, this is only true for dependent clauses (DC). With new independent clauses (IC) the subject is "reset", so you can't use an infinitive. The conjunctions "and" and "but" can prompt the emergence of a new IC or not. You know you have an IC if you can use a stop or semi-colon and the sentence is still sound. In the following sentence "and" isn't creating an IC:
I would like to sing and play ≠ I would like to sing; play.
But here it is:
I would like to sing and you would too = I would like to sing; you would too.
Now the same thing with "but":
I don't want to sing but play ≠ I don't want to sing; play (not an IC)
I don't want to sing but you do = I don't want to sing; you do (that's an IC)
In Spanish, there's only one translation for "and": y. Here are some examples:
Me gustaría cantar y jugar ≠ Me gustaría cantar; jugar (not an IC)
Me gustaría cantar y a ti también = Me gustaría cantar; a ti también (that's an IC)
However, there are two possible translations for "but": pero, that generally prompts an IC, and sino, that generally doesn't:
No quiero cantar sino jugar ≠ No quiero cantar; jugar (not an IC)
No quiero cantar pero tú sí quieres = No quiero cantar; tú sí quieres. (that's an IC)
Bottoom line: you can only use the infinitive in DCs that share the same logical subject with the Main Clause. Y can prompt an IC, pero almost always does, and sino hardly ever does. If you are dealing with ICs the fact they share the same subject is unimportant — you have to conjugate both verbs anyway:
Mañana iré al cine y luego iré a cortarme el pelo (two IC clauses that share the same subject)
Mañana no iré al cine pero sí iré a cortarme el pelo (two IC clauses that share the same subject)