Advantage. I rule that advantage and disadvantage cancel each other out on a one-to-one basis--so a character who is prone and restrained has disadvantage (two sources of disadvantage) still has disadvantage on an attack roll even if an ally has taken the Help action (one source of advantage).
Bonus Action Spells. The PHB states that if you cast a spell as a bonus action, you may not cast any other spells of first level or higher on your turn. This rule strikes me as arbitrary and confusing, and is often misunderstood or forgotten about. Consequently, I don't use it: you may cast as many 1st-level and higher spells on your turn as you have actions or bonus actions for. Go wild.
Charm Spells. I generally rule that, for spells like Friends, Charm Person and Suggestion, keeping the target unaware that they have been enchanted is part of the spell's function. However, anyone not affected by the spell will be able to notice your spellcasting as normal (unless you can cast the spell without components). So, if you cast Suggestion on a guard, they will not notice you casting a spell--but another guard nearby might.
Counterspells & Components. You may only react to a spell (typically with a counterspell) if you are aware that it is being cast before it is completed. Usually, this is because spells have components: if you see someone chanting and holding out a focus, you can tell a spell is being cast and can counterspell it. Spells cast without components (such as with the Subtle Spell metamagic) cannot be observed in this way, and are consequently much harder to counterspell--but if you have another way to know a spell is being cast (such as a Detect Thoughts spell reading the caster's surface thoughts) you can still counter the spell.
Counterspells & Spell Identification. When a creature casts a spell, you may use your reaction to make an Arcana check and attempt to identify the spell, as per the rules in Xanathar's Guide. If, after you roll your Arcana check (succeed or fail), you wish to attempt to counter the spell, you may do so as part of the same reaction you used to identify the spell. (Of course, if you use your reaction to identify the spell and decide that it isn't worth countering, your reaction is still expended.)
Creature Knowledge Checks. On your turn (no action), you may make a roll to recall or intuit relevant information about an enemy. The skill used (Arcana, History, Nature, or Religion) varies based on the creature type. Insight can also be used versus humanoids. The DC of the check is determined by the creature's obscurity or the frequency of legends and false information about them (examples below).
DC 10 - ankhegs, goblins, dire wolves
DC 15 - hellhounds, hill giants, ghasts
DC 20 - true dragons, beholders, doppelgangers
DC 25 - aboleths, gith, vampires
DC 30 - demigods, the anteconflagrant lich Heidelach, the Tarrasque
If you succeed, you may ask one question about the creature (including its statblock). A few 'stock' questions include attacks, defenses (hit points and armor class), special attacks (offensive abilities), special defenses (resistances/immunities/other defensive features), or movement (types and speed), but you might also ask a different question about a creature's stat block, or about a creature's attitude, motives, or likely tactics. If you exceed the DC by 5 or more, you may ask one additional question; or two questions if you exceed it by ten, and so on.
Illusions & Enchantments. These spells tend to be somewhat DM-dependent. I tend to rule rather permissively on them, because I think that they can be used in some very creative and interesting ways. However, limitation stated in spell descriptions absolutely still apply--for instance, Minor Illusion can only create images of still objects, and Suggestion requires a suggestion that sounds reasonable.
Investigation vs. Perception. I rule that Investigation is for thorough searches made with your hands, and Perception is for observation without touching. Frequently--such as if you're searching for traps--either Investigation or Perception can be used; Investigation is typically slower and usually has a significantly lower DC.
Sprinting. If you do nothing on your turn except run in a straight line, you can move up to four times your movement speed. Attacks against you have advantage until the start of your next turn, and you have disadvantage on perception checks until then.