I'm playing a Bard. This question is for a Bard or a Wizard character.
Does my familiar's "Help Action" help with my spellcasting? For example with Detect Thoughts, or Charm Person, or Sleep, Bestow Curse, or a Fireball, or any spell attack like maybe Fire Bolt?
Short answer - yes it can help, but only with touch spells and/or spells with an attack roll.
Long answer is a little too broad, but I think it is important in the context.
First and foremost, you get familiar either from the Find Familiar spell (a Wizard's one), or from a class feature (like Warlock's Pact of Chain). Normally a 1-st level Bard cannot have a familiar, unless you get one of the specific feats. You might get it at 10th level though, with the "Magical Secrets" Bard feature.
You can find a familiar description in the Player's Handbook, page 240:
Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn.
It says the familiar "acts independently", so your character have no direct control over it (they can give it orders tho). You should discuss it with the DM to what extend you will directly control the familiar as a player.
Usually, being a Wizard, you get the following advantages from the familiar:
PHB states that a familiar "can take other actions as normal":
A familiar can't attack, but it can take other actions as normal.
This can include any efforts of helping your allies. If it is a familiar, is irrelevant in this case. Any creature can help its allies.
PHB describes two usual cases.
You can lend your aid to another creature in the completion of a task. When you take the Help action, the creature you aid gains advantage on the next ability check it makes to perform the task you are helping with, provided that it makes the check before the start of your next turn.
There are restrictions though, see more in the "Working together" chapter.
... you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally’s attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.
Emphasis mine - the familiar shouldn't just "be within 5 feet" and take a unclear "help action" in order to magically give you an advantage. The familiar do things - it shouts, fusses around, meddles with the enemy, pours out sand over enemy's head, you name it. Mechanically, it's always the DM who ultimately decides if there was an advantage.
A familiar (and most of the creatures) isn't a mechanical robot, capable only of several pre-defined actions. You character can do other actions as well. The DM says, what check you should make (if any):
When you describe an action not detailed elsewhere in the rules, the GM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.
What a familiar could do, for example:
Familiars can use the "Help" action in combat, pestering an enemy and distracting them.
Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally’s attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage. -PHB p.192
Thus, if a familiar distracted an enemy, and you were the next person to target that enemy with a spell which used an attack roll (Such as Scorching Ray, but not a spell like Sacred Flame which doesn't use an attack roll), you'd get advantage on the roll.
Also, familiars could "help" (no capitalization because it's not using the Help Action) with a touch range spell (like invisibility) by delivering the spell themselves, acting as a conduit through which your power flows.
when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. -PHB p.240
However, the "Help Action" can't assist you in any other way with spellcasting. To see why, note the text on "Working Together" (emphasis mine):
In combat, this requires the Help action (see chapter 9). A character can only provide help if the task is one that he or she could attempt alone. -PHB p175
Familiars can be useful in the ways outlined above in casting spells, but they cannot cast a spell "alone". If your Bard character was knocked unconscious, your familiar would have no chance of successfully casting Fireball, or Sleep, or any other spell. Thus, the familiar cannot use the "Help" action to enhance your ability to cast a spell, since they wouldn't know what they were doing.
NOTE: You might ask why a familiar is able to "Help" with your attack when they can't attack themselves. This is an example of the specific over the general: the "Help" action description specifically states that it can give advantage in combat to the next attacker, so that is what it can do, whether or not the person using "Help" can attack themselves. But in the absence of a specific rule about using the "Help" action to assist with spellcasting, we default to the general rule that says you can't use the Help Action on a task that you can't possibly do alone.
2nd NOTE: Some familiars can, indeed, cast spells. Familiars gained by the Warlocks Pact of the Chain may have spellcasting abilities, for example. But as you asked about Wizards and Bards, the familiars you have access to will not be able to cast spells, and thus will not be able to "Help" with them.
The Help Action cannot be used to assisst with casting a spell. The Help Action can for example be used if you want to help your Barbar to keep a door closed, if you want to help your Ranger to smash some pots, hold something while your Wizard wants to write something down or any other circumstance where you could reasonably help someone with a task.
It has no direct influence on casting spells.
Only you can cast your spells and your familiar has no influence on you casting any spell. It cannot cast your spells after all, it cannot use verbal commands to cast a spell, it cannot use the material components for you and it also cannot make the gestures you need to do with your hands. All of these are examples of what a spell might require you alone to do.
See the description of the Help Action for further details and a usecase for your scenario (emphasis mine):
Help
You can lend your aid to another creature in the completion of a task. When you take the Help action, the creature you aid gains advantage on the next ability check it makes to perform the task you are helping with, provided that it makes the check before the start of your next turn.
Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally’s attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.
This means that your familiar cannot directly help you with an attack. But: it can help you by feinting and distracting opponents near your familiar. If you make an attack roll against the distracted target you can do so with advantage.
This means it applies only to spells that require an attack roll, such as a Fire Bolt. Not to spells requiring a Saving Throw, such as a Fireball or spells that instantly hit like Magic Missile.
If the GM rules that it is reasonable that the familiar assist you in spellcasting (which most GMs familiar with the milieu will-- familiars are definitionally magical assistants), then the familiar can, of course, use the Help action, since that isn't the Attack action.
The Help action, however, merely provides you with advantage on the next ability check you make as part of the spellcasting, which for spells like Fireball means nothing. Even Concentration checks are actually Constitution Saving Throws, not ability checks. That said you are being helped, just not in any way that yields a mechanical benefit, and if you did end up needing an ability check related to the spell, you would have advantage on it. Dispel Magic, for example, benefits from Familiar assistance: the spell requires you to make an ability check to dispel spells of 4th level or higher, so you would make the first such roll after your familiar took the Help action to assist you with advantage, provided you hadn't made any ability checks as part of that task yet.
The familiar can also use the help action to grant you advantage on a spell attack against a target within 5 feet, but that's sort of an edge case, as it works because it's an attack not because it's a spell.
Lastly, a familiar can aid your spellcasting in ways not covered by the help action, for example by fetching you material components, aiding in the deciphering of spell scrolls you wish to add to your spellbook, sharing with you the location and manner of protection of artifacts that would aid you in your quest for power, teaching you the mystical traditions of its people, and interceding with its superiors on your behalf to gain you access to libraries of magical lore, markets with useful and hard-to-acquire goods, and the academies of Heaven/Faerie/Hell. Basically, there are lots of other ways beyond direct immediate power buffs that familiars can aid you in your pursuit of spellcasting power.
