They can have multiple senses. You can distinguish to some extent by their accompanying phrases, but the tricky part is knowing when you have a phrasal verb rather than a verb followed by a preposition.
go on has five separate senses noted in Princeton WordNet:
S: (v) continue, go on, proceed, go along, keep (continue a certain
state, condition, or activity) "Keep on working!"; "We continued to
work into the night"; "Keep smiling"; "We went on working until well
S: (v) happen, hap, go on, pass off, occur, pass, fall
out, come about, take place (come to pass) "What is happening?"; "The
meeting took place off without an incidence"; "Nothing occurred that
S: (v) advance, progress, pass on, move on, march
on, go on (move forward, also in the metaphorical sense) "Time marches
S: (v) continue, go on, carry on, proceed (continue talking) "'I
know it's hard', he continued, 'but there is no choice'"; "carry
on--pretend we are not in the room"
S: (v) go on, come up, come on
(start running, functioning, or operating) "the lights went on"; "the
computer came up"
The first sense takes a complement verb phrase in its present participle form. The remaining senses second, third and fourth senses are intransitive. The example that another answer gives ("go on to bigger and better things") is debatable whether it's better analyzed as a phrasal verb ("go on") + a prepositional phrase ("to bigger...") or if it's a case of preposition doubling akin to ("climb up to the fourth floor").
go ahead as a phrasal verb has only one sense listed on WordNet:
S: (v) go ahead, plow ahead (proceed (with a plan of action)) "He went
ahead with the project"
One sense that I can think of that's not phrasal verb use is would be like "you go ahead, we'll catch up", where "ahead" functions similar to "home" in "go home".