Which game/mode would you recommend a new player?

IN Easy... because it's, well, easy.

Start from the bottom, work your way up, I say.

Playing the games in order accomplishes nothing, I think starting with the simplest ones is best. IN is a really good first game because there are no gimmicks related to resources and unlike in EoSD the hitbox is visible. PCB and MoF are also good starting games; PCB was my first game. As for difficulty, there's no reason to play easy once you're used to the controls. Easy is by no means a prerequisite for normal, so skipping it altogether might be a better plan. I would also recommend any beginner to use a homing shot (or Yuyuko's spread in IN).

I wouldn't suggest playing them in order, but EoSD I feel is the simplest. For the hitbox concern... give them the hitbox patch?

I agree PCB and MoF are also good starting games.

I would likely have not continued to play Touhou if I started with IN, and I'm glad no-one was around to recommend it to me. It's easy, yes, but I find a lot of that is due to how many resources you get given; especially if you then give yourself more lives than default. Blasting through IN with loads of resources and large deathbomb windows is great if you just want the easy gratification, but the same can't be said for any of the other games in the series, or difficulties higher than normal in IN, so it's quite unrepresentative of the rest of the games.

Homing shot/Yuyuko's spread is only good up to a certain point, and also depends on the game e.g. the homing shot in EoSD is fairly bad for beginners.

I would probably recommend working on PCB for newer players. Probably some bias here since this was where I started at. PCB is pretty basic for survival and the cherry system gives the player a choice between risk and reward during a border. The game is also pretty generous with resources.

As far as difficulty is concern, I would suggest trying normal before you decide to go easy. I'll at least admit that I don't touch easy that often, but I don't think easy would provide enough for a newer player to work with compared to normal(I can only go by IN and LLS easy from what I can remember. Memory too hazy for PCB easy and I can't say much for the other games). Also, EoSD and LLS won't let you challenge stage 6 if you play on easy so it could feel like an incomplete experience with those games (I believe it was EoSD and LLS).

I guess another point I'll add even though it wasn't ask is to play with default resources even if the game gives you the option to play with more resources. More of a personal thing since I don't like seeing the player penalty message at the end of a stage and I remember I felt like I was playing worse with the added resources back when I was learning PCB lunatic. I don't think relying on those extra lives will help someone learn the game compared to working with the default amount of resources but that is just my opinion.

I guess my alternative suggestion would be IN. IN does provide spell practice so that helps out when it comes to learning specific patterns. I'm kind of wondering if IN's deathbomb window is a good thing. It might be good if it motivates the player to not rely on it and learn when to bomb due to how you lose both time and 2 bombs from deathbombing (excluding the scenario of a normal bomb being used if you get hit).

Indeed she would, but I thight we were referring to homing shots we'd recommend newbies.

My suggested starting order is (all games on Easy):
IN -> DDC -> TD -> LLS -> MoF
Then one can play whatever. LLS may be skipped by some who are too lazy to set up a PC-98 emulator, however.

I would suggest being cautious with PCB for newbies despite its generosity because it's hardly any easier than Normal pattern-wise, making that a surprisingly daunting Easy mode. Also ignore those who tell you to play Normal 3 lives from the get-go. Start at the bottom then work your way up - but feel free to raise difficulty and/or lower starting lives if you're not feeling challenged.

PoFV is useful at any time if you want to improve your reactive dodging skills.

In regards to skipping difficulties and not messing with resources:

I personally do believe that the higher you play the better, to start of with atleast. Of course for people which want to get stuff done when they start I'd recommend normal mode, it indeed does teach the basics (moving, focus usage, bombing when needed, learning patterns, how the PoC works and whatnot actually.) While at the start being challenging, it does help to understand the basics which honestly is all you'll really need at the start. Easy mode is fine if you really don't get anywhere in normal which kind of is highly unlikely, even if you're just starting.

Personally I think that IN > MoF > EoSD > MS > TD is a good way to do it, IN is kind of easier which makes it simpler to get into, MoF learns stuff like streaming for a bit; resource management. EoSD learns that there's a solution for almost everything I guess, as well as that randomness can still be dodgeable; is probably good for reactive dodging. MS Kind of gives the feeling of the older games as well as it learns you that the PoC isn't a point you always get autocollect at in every shmup.

That's atleast how I see most things as, the reason why I recommend starting of high is because surely, you won't 1cc it right away or get a thing done in it per se in a short amount of time, but you're pretty much set to develop yourself into it until you can survive it consistently enough to get a 1cc, basically, you'll just develop faster because you aim higher; it challenges you to develop for that. I wouldn't recommend someone to start of at lunatic but I can very well imagine that some people would be capable of growing into that almost right away. normal is the mode I'd recommend for everyone to start out in, those who think they could add more spice probably should pick hard.

I started my Touhou career trying to beat ran. It took a long 7 months of on and off playing to do so, but it was so blissful when I finally got it.

I don't have a recommendation for starting per se. Just do whatever you feel like I guess. But don't be afraid to jump difficulties.

For the longest time I was only doing normal mode and extra mode and was afraid of Hard mode and above. I thought I would never be able to handle something like that until a friend of mine by the name of Gamercal suggested I experiment with hard mode. Since then, I had jumped to Hard mode, then Lunatic mode, and then LNB, and now I'm reaching levels of LNN/N and a bit of scoring.

One of my favorite things to have done was to try and beat all the games at the level I was at. Before I moved to Lunatic I cleared all games on hard. Then once I got to Lunatic I cleared all games on Lunatic. Then I went for all games perfect Extra and all games LNB.

But you don't have to play all games. If you want to specialize in a few that's fine too.

Counterpoint: people who fail Easy modes are more common than you may think. Dear Leader is one of them. Sure, she only like, plays IN once every few months, but that counts, doesn't it?

My point is, you guys may be overestimating the skill of a new player. Streaming, bombing or path-reading may not come as naturally for them as it does for us.
My other point is that Easy and 5/6/7 lives is not as much of a cakewalk as it is for us.
Try to place yourself on the shoes of the other player, who may not be as skilled - or as patient - as we are, and may feel discouraged by pushing themselves too much from the get-go.

The ones that would do something of that extent and survive all 7 months without giving up are a strict minority.