Spell Mechanics

Basics

Any spell that you earn outside of your regular player level ups (ex. You find a book or scroll that you learn a new spell from) do NOT count towards the total spells you can know for your class. In addition, many of your spellcasting Class Features (usually in the sections where you choose a Subclass/Archetype) will give you access to spells that don't count towards your total, so keep an eye out for those.
Spellcasting Components: All spells have requirements for how they are cast. Some are Verbal (cannot be cast if you are gagged) and/or Somatic (requiring physical hand or other motions to cast, so they cannot be cast if your hands/feet are bound), and some require Material components. If your spellcasting class provides you with an spellcasting focus, this will forgoe the Material components required to cast the spell, unless the Material components are to be of a certain monetary value (ex. if a spell requires a gemstone valued at 200 gp). If these spells are used with a spellcasting focus, it will consume the necessary gp to cast the spell. Players who do NOT already carry a spellcasting focus (such as Rangers), must cast these spells with the specified components, or take the time to craft their own spellcasting focus.
Spellsave DC: For EVERY spellcaster class, your spellsave DC is equal to 8 + proficiency bonus + spellcasting ability modifier (Intelligence, Wisdom, or Charisma). This will be used as the DC for your opponents to beat if they are caught in the effects of your spell, and will be specified in the text of the spell. It can also be used as the DC for magical effects stemming from your use of magical items, such as weapons & herbs.
Spellcasting Attack Bonus: When the instructions of the spell direct the user to "make a ranged spell attack" or melee spell attack, or when otherwise directed by the DM, you will use this bonus for the attack itself, NOT the damage. The bonus is equal to your Spellcasting Ability modifier (Intelligence, Wisdom, or Charisma) + your proficiency bonus
Spellcasting Damage Bonus: Unless a specific damage modifier is codified in the instructions of the spell (ex. Magic Missile auto-hits its opponents and is given a specific damage modifier, to keep from being OP) or your class specifies otherwise, you do NOT get your spellcasting modifier added to the damage of your spells. However, some classes have features that allow you to add your damage modifier to certain spells.
Conjuration spells (homebrew rule): When casting a spell that conjures other beings, 5e RAW state that the DM will have the statblocks and lists of summonable monsters. While this is technically true, assembling all that on the fly is stressful and also slows down the game. Therefore, the following homebrew rules exist to manage spells like this.

1. It is ideally the CASTER’S responsibility to understand the mechanics of the spell, to look up which beings they can conjure, and to pick which ones to conjure. During the first few times the players cast these types of spells, the DM will assist in helping them apply the mechanics. However, we all have the internet and the players know ahead of time what spells they have, so they can easily look up a list of creatures they can summon and keep that as reference material. The DM can provide statblocks in game, but it would be preferred if the player looks that up themselves in-game or prior to the game.
2. Once announcing that you are casting the spell, you have 30 real-time seconds to decide what you wish to conjure. If the creatures have their own initiative counts, the player will roll their initiative as a group and run them when their turn is up. If the creatures go on the player’s initiative count, the player will run them as part of their turn. If the player who conjures the creatures leaves the game early in real life, the creatures will dissipate in game at the end of the round. 

Concentration: When casting concentration spells, the max duration will be specified. You can still peform other activities while concentrating on a spell, but the following activities will break concentration:

• Casting another spell that requires concentration
• Being incapacitated or killed
• Taking damage - have to succeed on a Constitution saving throw, DC is 10 or half the damage you took, whichever is higher. Must make save for each source of damage you take.

Spellcasting Focuses: Most spellcasting classes allow the caster to use some sort of focus to cast their spells (Druidic, arcane, holy symbol, etc.). These spellcasting focuses satisfy the material components of any spell, unless there is a monetary amount specified for the value of those components. If that is the case, you can expend the equivalent amount of gold to cast the spell through your focus
Component Pouches: A component pouch initially comes with all the material components you need for your spells that DON’T require a monetary cost in the spell description. Once a component is used, it would need to be gathered again before that spell can be recast
Dual Casting (homebrew rule): The Player’s Handbook indicates that you cannot cast a spell as a Bonus Action if you have already cast a spell as an Action that turn, or vice versa. However, in my games, you can do it if you've got the slots and the Actions/Bonus Actions needed.
Other spellcasting rules: While there will be leniency early in a given campaign or when first using a given spell, it is ultimately the player’s responsibility to understand all the rules, mechanics & limitations of their spells. The DM has a game to run and NOT look them up on the player’s behalf unless they believe the player is using it incorrectly, at which point they will direct the player to look up the rules. The DM will not bail the players out of a bad spellcasting decision unless there is a mutual misunderstanding of the rules, or if information is acquired narratively that would make casting the spell an out-of-character motion (DM’s discretion). If the player does not know their spellcasting attack modifier or their spell save DC (ex. Doesn’t have their sheet), they will take an attack bonus equal to their proficiency bonus & a spell save DC of 8+proficiency bonus.