It seems to me like I get 3 cantrips because I'm level 3 and 4 1st level spells and 5 2nd level spells. Is this right?
How many spells can a bard have at level 3?
If you look at the Bard Table in the players handbook or on DND Beyond you will see that it lists the number of Cantrips known, Spells known and Spell slots for each Bard level.
You will see that at Level 3 you should have:
In terms of the level of your spells, in the Spells Known section of the Bard Class Features you will see that:
Based on this, at 2nd level you will have 5 known 1st level spells. When you level up to 3rd level, you can (1) learn a new spell of 1st or 2nd level and (2) you can also replace one of you 1st level spells with a 2nd level spell if you wish.
Thus, you will know 6 spells, with at most 2 of them being 2nd level spells and the rest being 1st level spells.
You can find this information listed in the Player's Handbook in the table under the Bard class description.
If you read the line for level 3 you will see that a 3rd level Bard will know 6 spells and 2 cantrips. They also have 4 1st level spell slots and 2 2nd level spell slots with which to cast those spells.
As a 3rd level bard you have access to 2 cantrips and 6 known spells; also 4 1st level spell slots and 2 2nd level spell slots to cast those known spells. You had 5 known spells at 2nd level and got one more known spell at 3rd level. However, when you gain a level of bard, you can choose one bard spell you know and replace it with another bard spell from the bard spell list that you're able to cast at your new level. This spell can be of inferior, same or superior level than the spell you're swapping.