So I took the Magic Initiate feat and got myself a familiar. I want to use it in combat, usually performing the 'help' action to give my attack advantage but occasionally to deliver one of my Touch spells. The best description of how to do this I found is posted below. Does this make sense to you guys? Thanks!

If I Have First Initiative Before the Familiar:

On the caster's turn, he readies the spell (spending the spell slot) to be released when the familiar touches the target. Readying the spell uses the caster's concentration. If he takes damage, he must make a concentration check or lose the spell.On the familiar's turn, familiar moves to the target, this triggers the wizard's ready spell, the spell is released, the familiar uses its reaction to touch the target, the familiar can move away if it has any additional movement.

If the Familiar Has First Initiative

On the familiar's turn, the familiar moves next to the target.
On the caster's turn, the wizard casts the spell, the familiar uses its reaction to touch the target. This doesn't use the caster's concentration. Since this is the caster's turn, the familiar does not get a move until its next turn.

I believe the first case isn't doable, at least if you go stricly by the rules. The wording for delivering a touch spell is:

Finally, 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.

The moment you cast a spell is the moment you take the ready action, as we can read in the PHB:

When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs.

Releasing the spell is not a valid trigger for the familiar, only casting is.

The second case is fine. The familiar could even take the dodge action to potentially avoid harm, since it's not using that action unless a dash was needed to close the distance. By the same token, if you have a familiar with the flyby feature, like an owl, you could also ready movement with the familiar to fly away as soon as the spell has been delivered, without provoking an opportunity attack.

Thanks for taking a look! I see what you mean about the first scenario. I'm hoping my DM will allow the release to be a trigger. I ran this by him pre-game, but he was vague about it, so will only find out if I try it in combat.

In the second case, the familiar can have a reaction AND an action in the same turn? Sorry, I'm new to this...

Yes, reactions and actions are separate resources. A creature gets both its action and its reaction back at the start of its turn. A reaction can be spent on anyones turn (even a your own) if a trigger (such as your druid delivering a spell or an opportunity attack) happens.

Thanks!

Even though Druid's get find familiar at 2nd level, albeit for a limited amount of time, I think taking Magic Initiate to get it without limits is a great idea. Find familiar is my favorite spell in the game. The only limitation for a druid taking find familiar is that druids don't have any offensive touch spells that can be delivered by a familiar except Contagion (which I think is an oversight on the part of the designers), but Wizards do, of course. So, take them! :) And you can still have your familiar deliver defensive spells like Cure Wounds.

 I was looking forward to being able to deliver Plane Shift and banish in one shot - but I don't know how to get the fork attuned to the plane spell component and my DM says I have to figure that out 'in-world'. We're nowhere near a city so can't see if I can buy one. Lot of undead in my campaign and sending a biggie to the Positive plane of radiant energy sounded cool... but aside from that, and Contagion, as you cited, not much you can have it deliver damage-wise. Will have it perform the Help action in combat to give me an advantage on my attacks or to deliver healing to others. But there are so many other things you can do with it - I have an owl and the scouting ability alone is worth it.

Given that the RAW of Plane Shift doesn't explicitly say that the material component is magical, I'd think a Wizard would be able to create it using fabricate, which is only a 4th level spell. It's probably a DM decision and your DM doesn't sound all that player friendly lol, but it's worth asking. 

I don't really understand why you'd need to cast both Plane Shift and Banishment, wouldn't the former suffice? Maybe in case the target saves on one?

I meant, only cast Plane Shift, which does the banishing itself, not the spell Banishment. The component fork has to be 'attuned' to a plane of existence of your choice - not sure Fabricate would do that. DM's all seem to make up their own ways you can attune it. It's never easy. Hoping I can pay someone to do it someday. Would be an awesome spell to have in your arsenal.

Re casting plane shift alone (and not banishment), okay gotcha.

Re the obtaining the tuning fork, I wish there were more official guidance about how to obtain the tuning fork rather than just leaving it up to a DM. IMO the 2 main options are obtaining it through an adventure or obtaining it from someone (of a class) who can cast the spell. 

Yeah, wish there was an official way to do it also. My party is not going to want to quest to get me a spell component. And from what I've read, attuning it can't be done through a spell. The fork needs familiarity with the Plane it's being attuned to. Barring going to the plane with the fork, you may do something else a DM comes up with, say leaving it with some creatures from that plane for 24hrs and seeping it in their blood, or simply having the fork forged from something metal that is from that Plane...

Re: delivering touch spells with a familiar. As it turns out, aside from Plane Shift, druids don't have any worthwhile damage-dealing spells that are Touch. Wish I had figured that out before I got the familiar. There are some healing touch spells you could use it to deliver during combat. But in my case, while we're playing, I can never tell who needs healing. We're high-level, they have ways of healing themselves, and their HP don't show on the VTT... so not something I'm likely to do.

Also disappointed to find out that the 'advantage' you get from your familiar performing the 'help' action during combat only applies to spells that require an attack roll, not a savings throw. Druids have 9, to be precise, mostly cantrips, that at in high level combat, don't deal enough damage to be useful. Beginning to think I may not have enough worthwhile things for the familiar to do in combat at all. :(

How rare or common planar travel is depends your campaign. In my current campaign, it's common--we just fought a BBEG Rakshasha and sent him back to his home plane when he hit 0 HP! In any of the bigger cities in my campaign, it would be easy to find someone with knowledge of, or experience traveling, the planes; it would also be easy to find a magic shop that had someone who had or could get me (say, within 1 weeks time) an item from other plane. Cost would vary depending on what the item is.

I'm also disappointed about the lack of offensive druid touch spells, esp since Druid's get familiars since TCoE! I'd really like to see an offensive druid touch spell in the new PHB. Although I think find familiar is probably the best spell in the game, and I think Wizards are the most powerful class in the game, instead of taking the Wizard Magic Initiate feat for my druid, I took the Cleric Magic Initiate feat for a few reasons. (1) Druids get find familiar at 2nd level since Tasha's, albeit for a limited duration, (2) Druid's don't get any offensive melee spell attack touch spells that can be delivered by a familiar, and (3) Druids and Clerics have the same primarily spell ability (WIS), so I can be a druid and take Cleric offensive spells and get to use WIS as my spell attack modifier. Inflict Wounds is the best 1st or 2nd offensive spell in the game (3d10 damage) and my 1st level druid gets to use his +5 WIS spell modifier to hit. And as a cantrip Toll the Dead deals the most damage in the game (does 1d12 if the target has already taken damage). And I get to use +5 WIS bonus to the target's (Wisdom) saving throw. Druids are one of my fav caster classes in the game, but they lack single-target offensive spells, but taking the Cleric Magic Initiate feat fixes that. It's almost OP ;)

Here's my druid with the Cleric Magic Initiate feat: https://www.dndbeyond.com/characters/121977514/7xHrIS