Let me see what I can contribute. Quoting from your question,
When we were first starting the game, we calculated the attack roll by
1d20+weapon stats+ability (but we changed it to 1d20+ability); for
example, for how we are doing it currently, if I used a shortsword and
roll a 10 then 10-1=9. We used that to see if our attack hits or not.
This is almost correct. While I'm not sure what you mean by "weapon stats" (but see footnote), you are correct in surmising that you calculate attack roll (which determines whether you hit your target or not) with a roll of 1d20 + modifiers.
The modifiers which are relevant here are proficiency and ability. With any weapon with which your character is proficient, you can add your proficiency bonus (+2 for a first level character, rising as you gain experience) to your attack roll. Your ability bonus is typically strength for melee attacks, or dexterity for ranged attacks (with exceptions). In this case it is the modifier, which on a character sheet is represented by the little number under the ability score. (I.e., if you have 12 strength, your strength modifier is +1)
At the start we used weapon stats (5+1d6+3+proficiency bonus+-1+little
number (under the modifier) for damage. We didn't know what to do w/
damage so we went straight to hit points (goblin ~7 HP but rolled a
total of 10 and considered them dead.)
This is where I'm a little confused by your description. Assuming you hit with your attack roll, you roll damage; this will be some funny-sized die which depends on your weapon, plus modifiers. In the short-sword example, the damage die is 1d6. To this you add modifiers, which is typically just your ability modifier as described above. Other modifiers include bonuses from using a magic weapon (ranging, typically, from +1 to +3), some class features, and a variety of other sources. You do not, and I cannot stress this enough, add your proficiency to damage. So, a typical first-level character with 14 strength smacks a goblin with her short sword. After rolling to hit (and assuming a good hit), the attack will deal damage equal to 1d6 (short sword damage die) +2 (strength modifier).
Footnote: Sometimes, the text will shorthand everything for you. Be careful with this. The DM often has stat blocks for their NPCs, with everything pre-calculated; so for example it may list somewhere that the goblin's spear does 1d8+1 damage. This stat block should already account for everything described above. Computer-generated character sheets often do the same. Keep an eye out for this and make sure you don't double-dip! If unsure, do the math yourself. Add up your own attack bonus and see if it agrees with the stat block; start with the basic damage die and add modifiers, and again, see if it agrees with the stat block.