How do I calculate if a spell hits and how much damage it does?
I'm not entirely sure I understand what the PHB is telling me about spell attacks.
Say I'm a wizard who's casting Fire Bolt (I could also be a Cleric casting Inflict Wounds or what-have-you), the spell itself doesn't say anything about making an attack roll or the enemy making a saving throw.
So do I make an attack roll and then add in my INT mod + proficiency bonus? Or does the target have to make a saving throw against a DC of 8+ my INT mod + proficiency bonus?
And, assuming I hit, do I add my INT mod + proficiency bonus to the damage?
Proficiency Bonus is not added to damage rolls.
Proficiency bonus is added to attack rolls, ability checks, and saving throws for which you are proficient. Damage rolls will add the appropriate modifier for the stat which was used to make the attack. However many spells, especially cantrips, do not add your casting mod to the damage.
There are some class abilities that allow proficiency to be added to damage rolls, but those are a specific exception rather than the general rule.
Firebolt requires you roll 1d20+INT+proficiency vs. AC and deals 1d10 damage
Make a ranged spell attack against the target. On a hit, the target takes 1d10 fire damage. A flammable object hit by this spell ignites if it isn't being worn or carried. - PHB p. 242
“Spell attack” = roll a d20 and add prof and mod. Spell save means target rolls defense.
Each spell's description indicates whether you need to make an attack roll, or the target needs to make a saving throw, or both.
Many spells specify that a target can make a saving throw to avoid some or all of a spell’s effects. The spell specifies the ability that the target uses for the save and what happens on a success or failure. The DC to resist one of your spells equals 8 + your spellcasting ability modifier + your proficiency bonus + any special modifiers.
Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus. Most spells that require attack rolls involve ranged attacks. Remember that you have disadvantage on a ranged attack roll if you are within 5 feet of a hostile creature that can see you and that isn’t incapacitated (see chapter 9). - PHB p. 205
There are two types of attack spells requiring d20 rolls: melee attack spells and ranged attack spells.
The difference is that melee attack spells will not be made at disadvantage when next to an enemy combatant whereas ranged spells would be (and all ranged attacks for that matter).