My character is a bard and I'm planning on taking a level in Warlock. Do I get the 2 cantrips and 2 spells from the Warlock spell list that a new Warlock character would get in addition to the Bard cantrips and spells that I already have access to?
Yes, when you multiclass into Warlock you will gain 2 cantrips and 2 spells known
When you multiclass:
When you gain a new level in a class, you get its features for that level.
Pact Magic is not quite the same as Spellcasting, but is a class feature which gives you 'facility with spells':
Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
So, when you gain your first level in Warlock you'll get two Warlock cantrips and learn two first level Warlock spells, just as a single classed level one Warlock would.
If you're uncertain how your Warlock spell slots, gained from Pact Magic, and your Bard spell slots, gained from Spellcasting, interact with each other please refer to the multiclassing rules (linked above) for more information, or take a look at this question unpacking a comparable case. If you get confused you can open a new question.
You won't gain any new proficiencies when multiclassing into Warlock, as the only proficiencies which you could gain, light armour and simple weapons, you already have as a Bard.
Also, remember that 'if a cantrip of yours increases in power at higher levels, the increase is based on your character level, not your level in a particular class'. So any cantrips you have, whether from Warlock or Bard class features, will increase in damage as soon as you reach a total character level of 5. You won't need to wait for them to scale until you reach level 5 in one of your classes on its own.