The academic English for a simple sentence construction is subject, verb, object. So, "I" is the subject, "like" is the verb, and " reading" is the object.
If you are doing the home work for your school, then " I like to read " is an incorrect answer. You should choose I like reading. However, "to read" is an act, you like to read that means you like to do something, it is reading.
If you are asking me for opinion, I agree with you that you can say I like to read instead of I like reading. Precisely, these two sentences may not have the same meaning. For example:
Q: Are you working hard? A: I am hardly working.
Hardly is an adverb which modify working, grammatically, it is correct. But its meaning has been changed, it is not what the question asked.