Last night my brain screamed at me, “No more Arial!!!”
I’m sure this is an enormous topic in and of itself, but in general, are there some fonts for Headings and Body Text that look way better than your standard Arial and Times New Roman stuff?
(**Note: I want a font that will work on nearly everyone’s computer or devise, so I don’t want some fancy font that no one has!!)
Take a look at the following two links; they’re are plenty to choose from.
There are numerous sites out there that will give you good advice on pairings/combinations for fonts that work well together. I wouldn’t describe fonts as my particular forte though, so other than point you in the right direction. You could do worse than check out the Typekit Gallery for ideas though.
If you are planning to use web safe fonts only then take a look at cssfontstack.com. It tells you which fonts are available on Mac and Windows platforms. Personally I’m a big fan of Georgia. It looks beautiful and it is a web safe font so you don’t have to worry about it being installed on others’ computers. Some other beautiful web safe fonts are: Palatino Linotype (Win: 99.29% Mac: 86.13%), Century Gothic (Win: 87.62% Mac: 53.15%), Gill Sans (Win: 58.54% Mac: 95.5%) and Copperplate (Win: 66.87% Mac: 92.61%). Personally I like Courier New too but it is a monospaced font so its style is not suitable for all websites.
That being said I think we all should take advantage of CSS capabilities like @font-face, it saves a lot of headache.
The trend now is to use web fonts, which can either be stored on your site and downloaded by the end user, or links to a 3rd party storage site. Make sure to check out @chrisofarabia’s link. I’m a fan of Google fonts. They are free and work really well.
Just to build on what @ralphm has said, I’m a big fan of the way medium.com does its typography - I believe they use Google fonts, though at the moment, I can’t find the article I read that said that. I’ll see if I can find it.
Yes, if you peek at the source code you often see a link to Google’s fonts in there. Likewise, their typography makes reading a pleasure.
I think the point I meant to make, though it didn’t come across well, is that the use of sites like Google fonts are very much a mainstream approach these days to dealing with typography. There isn’t the same requirement to ensure that you are using websafe fonts, or somehow validate whether the end-users’ local machine has the same font available as the one used to create a site. That’s not to say that one day one of these sites may not vanish in a a puff of ‘dot com bubble collapse’ smoke, but that’s why you declare a set of fall-back fonts.
And here on Linux, I have none of those. (I do have Georgia and Courier New.)
Old school… 
I wasn’t suggesting their use - just remarking that they’re the only two “web-safe” fonts on that list that I actually have installed.
I always have Georgia on my mind. It’s still may favorite font.
I quite like Georgia for headings myself. (I never use it for body text, because I find serif fonts more difficult to read than sans serif over blocks of text.)
As a designer Linux is a black box to me. Well at least you have Georgia. 
This list of tools might prove useful - http://designinstruct.com/roundups/font-pairing-resources/
What is @font-face?
So every time someone visits your site, they have to download all of the fonts before viewing your web page? That sounds horribly inefficient!
How important is choosing font?
From what I have heard, you could spend years learning all about font.
To create a nice, professional-looking website for my clients, do I need a Ph.D. in Typography?
You could spend years learning everything about anything.
I’ll take “what is caching” for 400 Alex.
Caching doesn’t make things miraculously appear in your browser - you still have the initial overhead.
Also, what if the website that has your website’s font is down?
Or if someone has poor internet connectivity which I sometimes do when I travel?
It seems that there would be fonts built into operating systems or browsers or however fonts work that might be safer.