Your question title asks about when to use 'verb + to infinitive' or 'verb + the -ing form'. This is a difficult aspect of grammar for English learners. There is no rule which will help you decide for each verb which complement to use. You simply have to learn case by case. The answers provided in the question linked to by Edwin Ashworth in his comment have useful lists of such verbs.
Now to the specific question about the use of feel. This is a different issue because it follows the pattern:
copula + adjective + verb
In most cases the verb is in the to-infinitive form:
But this doesn't work for feel:
?I feel better to know ... .
It needs to be "I feel better knowing ... ."
I have no rule here, but the difference seems to be that the verb feel in this context refers to my mental, emotional state. In the other examples, conversely, I am reporting on the impression made upon me by some other person or thing.
It is possible to conceive of the same examples above with I as the subject:
- I looked ready to cry.
- I seem hard to like.
- I sound too good to be true.
But again in these cases the focus is not on my mental state but on the impression I give others.
It is interesting to note that the to-infinitive does work with the impersonal subject:
It simply feels better to know that a pen is handy.
It is also interesting that feel + the to-infinitive works in the following example:
I feel sorry to be such a nuisance.
There may be an explanation as to why this is permissible, whereas
I simply feel better to know a pen is handy
is questionable. But no explanation occurs to me at the present.