My school is on a quarter system so the calc series is divided into 4 quarters. I'm currently in my 3rd quarter of calculus which covers:

This may sound like a dumb question but, should I take the 4th quarter of calculus next called "Vector Analysis"? Is this Calc III? It covers:

Or should I take a linear algebra course that covers:

When should I take elementary differential equations?

Finally after what course should I take Advanced Calculus? It covers:

Which courses are good to take concurrently, and which ones are good to take before one another? I am a first year math/statistics major and could really use your help. Thank you!

What is usually "calc 3", is split between the second half of the third quarter and the fourth quarter of calculus at your school, according to what you posted.
calculus 3 = multivariable = partial derivatives, multiple integrals, and vector analysis.

Since you're a math major you'll have to eventually take all of the courses you mentioned; physics students could probably skip advanced calc as it seems pretty rigorous.

Linear algebra and multivariable are closely related (from personal experience, multivariable makes a whole lot of more sense when you know linear algebra) I suggest you take both next quarter.

Are you at UC Davis?
Vector Analysis
Advanced Calculus
linear algebra
can be taken in any order.
elementary differential equations should be after linear algebra
Math students should take the other better linear algebra, the one you mention is for other majors
two things to keep in mind
1)Advanced Calculus and linear algebra are transition classes they are a bit different than earlier courses and may or may not require adjustment
2)the biggest issue is scheduling, courses are not offered every term so you need a plan ahead so future classes are neither delayed nor bunched up

This looks a lot like UCD math sequence. If so, then what lurflurf says is right. You can take that LA course, but when you take the one for majors it will only count for one unit and will be very similar with more theory it appears.

The other linear algebra class is called Modern Linear Algebra, and I understand it's proof-based. It covers:

I have never taken a proof based math class before, so should I take Advanced Calculus before taking this class?

Should I take the elementary differential equation class before modern linear algebra?

So then I'll probably take multivariable calculus with the elementary differential equations class, if offered in the same quarter, and then take advanced calculus. Then I should probably take modern linear algebra, right?

You say you're a math major. If you haven't taken a proof based math course yet then you've never seen math, meaning math as a mathematician thinks of it. What you've seen is math as a tool, as a scientist or engineer thinks of it.

So if you are actually thinking of yourself as a math major, you should take a proof based math class as soon as possible. You should self study something that's mathy and rigorous as soon as you can: maybe number theory or logic and set theory. Figure out if you like it, if you're good at it. Do it over the summer.

If proofs worry you
Advanced Calculus
Modern Linear Algebra
Introduction to Abstract Mathematics (a really bad class)
Convex Geometry
Number Theory
Euclidean Geometry
Combinatorics
are the transitions classes. These involve proofs, but are a (hopefully) gentle introduction. Modern linear algebra is probably good to take first, but it varies from person to person. You could take linear algebra and modern linear algebra, but it is probably not worth the trouble. Elementary differential equations uses linear algebra so unless you know some from somewhere take either linear algebra class first. So again take
Vector Analysis
Advanced Calculus
modern linear algebra
in any order
elementary differential equations after some linear algebra

I take it you are talking about fall?
I do not know where you are at in statistics and general ed, but you want something like

fall
25
67
other math class
general/stats
winter
21d
125a
other math class
general/stats
spring
22b
125b
other math class
general/stats

It would be fine also to push 125 to third year (some people find it hard), but it might jam you up or limit your options. Some people like to not take 125 and 150 at the same time.