Substituting patterns in expansions can be done with the "${myvar/pattern/replacement/}" form:

bash$ promise='I'\''ll do it today.'
bash$ printf '%s\n' "${promise/today/tomorrow}"

This only replaces the first instance of the pattern. If you want to replace all instances of the pattern, use two slashes before the first pattern rather than just one:

bash$ promise='Yes, today. I'\''ll do it today.'
bash$ printf '%s\n' "${promise/today/tomorrow}"
Yes, tomorrow. I'll do it today.
bash$ printf '%s\n' "${promise//today/tomorrow}"
Yes, tomorrow. I'll do it tomorrow.

Note that the pattern being replaced uses the wildcard (globbing) syntax, with * matching multiple characters, and ? matching only one; it is not a regular expression as might be used with sed or awk:

bash$ promise='Yes, today. I'\''ll do it today.'
bash$ printf '%s\n' "${promise/today*/I\'ll do it soon.}"
Yes, I'll do it soon.