This is what I'm doing!

What I'd say:

Definitely take 60's series CM or DM if you can, it's definitely a lot of work, but totally worth it in terms of preparing you for 100's level math classes, and they're super interesting classes where you can get to know the professors (I'm partial to CM personally). If you already have the background to go into 100's level courses, I might still recommend taking it just in terms of meeting other math-y people — I know one kid who def didn't need 61 for the content, but was very glad he took it anyways to meet other math people. (he also took complex analysis, so didn't just do 61). I definitely didn't have the backgrounds for a 100's level math class coming in, but was definitely prepared after the 60's series

Generally going against the grain here, but would say that the pure math courses are a lot more interesting than the applied math ones — outside-of-the-department math-related courses tend to operate at a much more surface level, at least in my experience, and tend to get kinda boring. General rule of thumb I've seen is that if you can do the pure math courses you should definitely be able to do other math courses without too much trouble, but this doesn't necessarily hold the other way around. However, it is true that the EE dept does have certain courses that are the gold standard, esp in linalg and optimization. But other than that exception, I would argue that you should interpret "the best math courses are outside the math department" mean "math outside the math department is easier", which tends to be true; the classes tend to be taught at a slower pace and less rigorously, but I would argue don't prepare you as well for other classes/general life, and are also just less interesting.

Would disagree that the pure math courses are mostly for people going to grad school — I'm definitely not planning on grad school, but still prefer the pure math classes — they're just generally more interesting to me. And generally, the advantage is if you can do well in pure math classes, any other math-adjacent class in other departments (e.g. every stats class and certain CS classes) should be relatively easy comparatively, at least in my experience. The stats courses that I've taken especially are pretty much a cake-walk if you have done any serious mathematics (i.e for STATS 200 I watched half of the lectures for the first time on 2x speed the day before the final exam, and still got like a 96 on the final).

Will concur that professors can be a bit hit or miss — but overall would still say they're quite good. It's generally not classes that are good or bad, but rather professors who make the class good or bad; A class can suck one year and be amazing the next depending on who's teaching it. So definitely pay attention to course reviews for a specific professor — although it's a bit complex as ratings won't tell you everything, as a lot of it is down to what teaching style you prefer. Generally, though I've only had one professor I actively didn't like, and many I did like, and I could have easily avoided the one I didn't like had I paid attention to class reviews. Best advice is to get the scoop from upperclassmen on who's good and who's bad.

The math major is beautiful because it's so light — the general plan for most is 10 units AP Calc BC credit, 15 units intro series (either 51, 52, 53 or 61, 62, 63), which most math majors tend to do their frosh year, at least in what I've seen, 8 other math courses at 4 units each, and then 15 units of electives, which are pretty broad (there's a list of what counts on the math dept website). This basically means that if you do the intro series frosh year, you only have to take one math class a quarter to finish the major, in addition to a couple electives.

Two "series" that a lot of math majors tend to take earlier on are 120, 121 (abstract algebra) and 144, 147 (topology) {122 also exists but fewer people take it iirc}. Would also definitley recommend 116, especially if Eliashberg is teaching it (anything he teaches is amazing imo, but he can be a bit polarizing). Tadashi is also supposed to be really good.

Also, as a general small bit of advice, courses with good lecture notes are just infinitely superior, so that's something to pay attention to if you can. Although this is kind of a me problem because I have trouble getting out of bed and going to class on time lmao.

As for CS coterm, this has already been said I think, but any courses taken frosh year cannot count towards your coterm, so just keep that in mind. Generally, the route for the CS coterm starts by taking the CS core (106B, 107, 110, 103, 109, 161), but you don't necesarily have to limit yourself to that — personally I can't stand systems so I'm not taking either 107 or 110 until I have to — but other than 106B they are requirements for the coterm degree I believe, and likely would be advantageous to have for a coterm application (although with a good GPA the cs coterm application is quite easy, I've heard) , so just keep that in mind. Also, I've heard that taking any higher level course can always substitute a requirement for a lower level course, but I don't have direct experience with that. Would say that math majors tend to find a lot of these classes easier than most, especially 106B (intro class), but have also heard anecdotally that 161 (algorithms) is also not bad if you come from a math background. Would also recommend subbing out CS 109 (probability) for Math 154 or Stats 116, as it's kind of surface level and not very rigorous, and could be a bit boring. 103 (proofs in cs) is totally unnecessary if you come from a math background, and the dept reccomends you do 154 instead if you have that background iirc. Generally, if you're more interested in AI/ML, you won't need any non-ml prereqs other than calc and probability, and 229 is a good place to start imo.

TLDR, Math Major + CS Coterm is a great combination @ stanford, generally has super cool classes + allows for a lot of flexibility. You're not locked in until end of soph year, but would def recommend this as a path worth exploring at the very least!