You are correct to roll 4d6 and discard the lowest of the 4, as detailed on pages 12-13 of the Player's Handbook1. Add together the remaining three dice results, which will give you a number somewhere between 3 and 18. Repeat this process six times and then assign the numbers you get to the six abilities.
(Racial ability score increases may also apply, check under '[Race] Traits' in your race's description further in the PHB).
Once you have determined your ability scores, look at the table (also on page 13 of the PHB) to find out what your ability modifier is or calculate it yourself2, a number between -5 and +10 (these could be where you've seen negative numbers - it's important to know the distinction between your ability score and your ability modifier).
Specifically for your half-elf bard, once you've assigned the six random numbers from the dice, you should increase your Charisma score by 2, and two other ability scores of your choice by 1 each. You will also be able to increase ability scores at 4th, 8th, 12th, 16th and 19th levels.
- There are two other methods detailed on p13: you can use the scores 15, 14, 13, 12, 10, 8 (distributed as you please), or (at your DM's discretion) you can use a point-buy system to get specific scores.
- To calculate your ability modifier subtract 10 from the ability score, then divide by 2 and round down.
For example if your ability score is 17: floor((17-10)/2) = floor(7/2) = floor(3.5) = +3 modifier.
Or if your ability score is 9: floor((9-10)/2) = floor(-1/2) = floor(-0.5) = -1 modifier.