Sage Advice says...
After this question was asked, the 2016 Sage Advice Compendium has provided additional guidance that clarifies the rules from the PH.
Did my character actually perform a chant before giving the suggestion?
Yes, but the chanting is “so subtle” that “it typically goes unnoticed.”
Sage Advice Compendium 2016 (http://media.wizards.com/2016/downloads/dnd/SA-Compendium.pdf) states on page 15:
Verbal components are mystic words (PH, 203), not normal speech. The spell’s suggestion is an intelligible utterance that is separate from the verbal component.
Some have argued that the chanting of mystic words is always obvious, and would be easily identified by anyone present. But that’s not how the designers envision a spell like suggestion working — read on.
Do you always know when you’re under the effect of a spell? Would a bystander notice that a spell was being cast?
Not “typically.” Per the basic rules on spellcasting in the PH, whether a spell is obvious or not depends upon its effect:
An effect like crackling lightning is obvious, but a more subtle spell effect, such as an attempt to read a creature’s thoughts, typically goes unnoticed, unless the spell says otherwise.
PH, p. 204, Targets:
The spellcaster doesn’t have to use any skill or ability to conceal the casting of a spell with a subtle effect — typically.
While in my opnion those rules are cut-and-dried, there has been a fair deal of debate and confusion about this. Fortunately, we’ve gotten some additional guidance. Again from Sage Advice Compendium 2016:
Some spells are so subtle that you might not know you were ever under their effects. A prime example of that sort of spell is suggestion. Assuming you failed to notice the spellcaster casting the spell, you might simply remember the caster saying, “The treasure you’re looking for isn’t here. Go look for it in the room at the top of the next tower.” You failed your saving throw, and off you went to the other tower, thinking it was your idea to go there. You and your companions might deduce that you were beguiled if evidence of the spell is found.
There is wiggle room for interpretation around “failed to notice the spellcaster casting the spell,” and how difficult it would be to notice a spell being cast. But this examples draws a picture of an adventurer (not some orc or peasant) being suggested, and it’s not immediately obvious the target or his companions that magic was used against them. (And remember, the range of Suggestion is only thirty feet.)
How subtle is subtle?
We don’t get any rules for how hard it would be to detect a spell like suggetion, just some guidance on what kind of skills could be used:
It’s ultimately up to the DM whether you discover the presence of inconspicuous spells. Discovery usually comes through the use of skills like Arcana, Investigation, Insight, and Perception or through spells like detect magic.
Sage Advice Compendium 2016, p. 11
Thus, to detect a spell like Suggestion, at least some kind of skill check would be required.