you're overcomplicating things way too much...
I am using TextWrangler to create a new template to be used with Wordpress. What I'd like to know is if I set the basic text formatting in the main style.css file, will it translate to the Page/Post/Comment WYSIWYG editor.
In short "no" it will not.
The "style.css" file only controls how the published
page or
post will look like, not the WYSIWYG editor itself within the WordPress admin.
If you want to change the default WP admin CSS rules then, just as the poster
Hansr mentioned previously, you can do so
here.
But you really don't want to do that because:
- you're only creating extra and unnecessary work for yourself & potentially your client to manage and keep updated.
- You could potentially destabilize the WordPress install as a whole unless you're completely sure of what you're doing.
- the default WYSIWYG editor looks and works perfectly as it is.

Instead, what you need to get into the habit of doing is previewing the post/page edits using the "preview" button under the
publish heading in the WP editor as shown below:
Also, look at WP's info on
how to create & manage pages to get a better understanding of how to create page templates for themes.
... I have to manually switch to HTML view mode, and input the class tags in so that the paragraphs (ex: <p class="...">), etc., will format themselves to the appropriate settings.
This is normal and virtually the best way to implement your
specific CSS class rules within WP's admin content WYSIWYG editor. This what is called an "inline formatting" of a particular style rule to be applied within the page/post content area.
... I don't use too much custom formatting throughout the template, so I was wondering if I should just set the CSS text settings throughout the theme itself, and would that auto format things when I'm using the Page editor.
Yes, do it this way. But no, the Page editor will not show all the formatting rules declared within the
style.css file until you "preview" the live page.
If you don't want to declare any specific CSS
class="xxx" attributes within the body content for the page/post and just want to have, for example, the default "H1" tag
(or for any HTML formatting tag such as "H2", "H3", "P", "OL", "UL", etc...) to present a
particular but default text format style without needing to use a
class= attribute every time within the WYSIWYG editor then you must
declare all the relevant styles for each of the
HTML Formatting Tags you plan on using within the "style.css" file to have those CSS rules applied by default.
...I'm going to be handing the template over to my client. They want to be able to add several other pages by themselves in the future, so I need a way where they'll be able to add the text to the page, and not have to dig in the code to add custom classes. They'd simply enter a heading like Welcome, highlight it, and set it to H1 with bold. The issue I'm having is that the text that WP's page editor generates doesn't use custom tags like I do when making custom pages on my end, so their text won't get color formatting, or a custom selected font.
Two part answer:
1st bolded area - You accomplish this by setting the CSS style rules for all relevant
HTML Formatting Tags you want to pre-determine by default within the "style.css" file, as I already mentioned once before above. Then your client can simply just use the WYSIWYG editor within WP to create, format and preview their new page/post content.
2nd bolded area: I think you're simply approaching things in the wrong manner. You need to to first declare all the standard formatting style rules within the "style.css" file before you can see them implemented on the
published or
preview version of the page. If you're going to use
specific inline style rules then you must use the
class="" or
style="" tag attributes (via the HTML editor) when editing the page/post content. Othewise, if you're using the "Visual editor" panel you'll only see the default style rules for the common formatting tags,
but not fully until you "preview" the page/post after saving your changes. Once you've applied all the rules to the
style.css file, try reviewing your content this way instead:
- Select the "HTML editor" within the admin panel of the page/post to see the code side of the content you're adding/formatting and make your adjustments to the content.
- Then click the "preview" button after you've saved your changes to see what the page/post will look like live.
- repeat steps 1 & 2 until you get the desired results, then click "publish" to make the page go live.
Hope all of this helps, good luck!