No, your numbers are not correct.

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:

Cantrips known: 2
Spells known: 6
1st level spell slots: 4
2nd level spell slots: 2

In terms of the level of your spells, in the Spells Known section of the Bard Class Features you will see that:

When you level up you learn 1 new spell.
The new spell must be of a level that you have Spell Slots for.
In addition you can choose to replace one of your existing spells with a new one.

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.