How to apply color in Markdown?

TL;DR

Markdown doesn't support color but you can inline HTML inside Markdown, e.g.:

<span style="color:blue">some *blue* text</span>.

Longer answer

As the original/official syntax rules state (emphasis added):

Markdown’s syntax is intended for one purpose: to be used as a format for writing for the web.

Markdown is not a replacement for HTML, or even close to it. Its syntax is very small, corresponding only to a very small subset of HTML tags. The idea is not to create a syntax that makes it easier to insert HTML tags. In my opinion, HTML tags are already easy to insert. The idea for Markdown is to make it easy to read, write, and edit prose. HTML is a publishing format; Markdown is a writing format. Thus, Markdown’s formatting syntax only addresses issues that can be conveyed in plain text.

For any markup that is not covered by Markdown’s syntax, you simply use HTML itself.

As it is not a "publishing format," providing a way to color your text is out-of-scope for Markdown. That said, it is not impossible as you can include raw HTML (and HTML is a publishing format). For example, the following Markdown text (as suggested by @scoa in a comment):

Some Markdown text with <span style="color:blue">some *blue* text</span>.

Would result in the following HTML:

<p>Some Markdown text with <span style="color:blue">some <em>blue</em> text</span>.</p>

Now, StackOverflow (and probably GitHub) will strip the raw HTML out (as a security measure) so you lose the color here, but it should work on any standard Markdown implementation.

Another possibility is to use the non-standard Attribute Lists originally introduced by the Markuru implementation of Markdown and later adopted by a few others (there may be more, or slightly different implementations of the same idea, like div and span attributes in pandoc). In that case, you could assign a class to a paragraph or inline element, and then use CSS to define a color for a class. However, you absolutely must be using one of the few implementations which actually support the non-standard feature and your documents are no longer portable to other systems.

I have started using Markdown to post some of my documents to an internal web site for in-house users. It is an easy way to have a document shared but not able to be edited by the viewer.

So, this marking of text in color is “Great”. I have use several like this and works wonderful.

<span style="color:blue">some *This is Blue italic.* text</span>

Turns into This is Blue italic.

And

<span style="color:red">some **This is Red Bold.** text</span>

Turns into This is Red Bold.

I love the flexibility and ease of use.

When you want to use pure Markdown (without nested HTML), you can use Emojis to draw attention to some fragment of the file, i.e. ⚠️WARNING⚠️, IMPORTANT❗ or NEW.