The manual uses Gotham (usually bold) for the sans serif titles and callouts, and Garamond for the main text. The example gameplay passages use GFY Thornesmith. Evil Hat have their own font for the Fate action glyphs.
Gotham is a priced font, but I hear Google's Montserrat is a good substitute and uses the SIL Open Font License. Garamond has priced variants (the manual uses Adobe's I think), but URW make a version called Garamond No. 8 — you can download a TrueType Garamond No. 8 from Github (under the AFPL). GFY Thornesmith is also priced, and I don't know of a good (free) visual substitute, but Blambot make a bunch of comic fonts that are free for personal use (eg. Heavy Mettle, Nightwatcher).
As for replicating the style, all I can say is: pay close attention to the little visual cues we often take for granted. Note the 45° corners on the black callout boxes, and the drop shadow. Note the indent for, and horizontal lines above and below, the example text. Also there's frequent use of bolding to emphasise aspects and other mechanical details. In general, it's got a very high-contrast, solid black-and-white style. Compare it to, say, most WotC D&D material where everything is in subdued browns, reds and cream colours with a lot of texture.
There's a fair amount of artwork in there that breaks up the walls of text; if you can't make your own artwork, just do a Google image search (eg. I just searched for "[genre of game] vector artwork"). There's a lot of stuff that's free for personal use, but remember to respect the license if you publish it. I like to have it "enter the page" from the side or lower corners and wrap the text around it, you might prefer to feature it within the page proper. The manual uses artwork with a little bit of texture (which contrasts nicely with the heaviness of the text, and so doesn't really need captioning or framing); personally I find that solid vector based stuff works fine as well, as long as it's in moderation.
Finally, note that it's the structure of the book that dictates these elements. The text regularly switches between "main text," example gameplay, and helpful tips. Each of these elements has its own style. Putting random paragraphs in big white-on-black boxes, or switching to a hand-drawn-looking font for no particular reason will just be jarring. Text with the same font should be able to be pulled out by itself and still make sense (or at least give you a coherent set of excerpts).
There's a lot of merit to trying to reproduce your system's look and feel; it can really help reduce cognitive load for your players switching between the official material (such as the manual) and your own handouts. Having said that, if you get close but not close enough, you can set off a bit of dissonance as your players' expectations are subtly missed in ways they can't quite identify.
I already mentioned this, but you said you were doing this for personal use but thinking about publishing your material later. Even if you're publishing it for free, it probably won't fall under "personal use" any more, and you'll need to be careful to respect the licenses of any fonts and artwork you use. This may simply mean captions for crediting images, or it may mean thinking about a real license.
When I do stuff like this, I like to keep a text file alongside my material with a list of anything I downloaded and used, along with source URL and a quick note about the license. (You'll also appreciate this when you render a PDF, accidentally delete your source file, and then wonder where you got that picture of a planet from.)