To make flyers one usually uses at least 3 different apps. One DTP app for the layout, one app for pixel based images and one app for vector based images. Preferably all three apps should be able to handle CMYK color for professional printing and ideally ICC profiles for color management.
The best free professional DTP app you'll find is Scribus
https://www.scribus.net
Most other free apps won't handle CMYK color very well and you'll need that for printing your flyers at a professional offset printing store. With Scribus, you could use RGB images for layout and then export the final flyer to a CMYK PDF, that contains the proper ICC profile for the kind of paper you'll print on.
I said RGB, because most free image editors can't handle CMYK.
Use Tif images in 300 dpi resolution that you could set up and then export with GIMP
https://www.gimp.org
The only free image editor I know that can handle CMYK so far is Krita
https://krita.org
Last time I used Krita, it was in beta stage. I'm still missing some features there, but it's noteworthy.
For creating the vector artwork like logos and exporting them as EPS, you could use
https://inkscape.org
At this time it isn't CMYK ready and can't give you accurate color.
GIMP and Inkscape would work in conjunction with Scribus to get a result that is professionally printable, but would have some color shift due to unavoidable automatic conversion of RGB to CMYK color, based on ICC color profiles.
If you can spend money, go with an Adobe Creative Cloud subscription. You'll get all the professional tools that you need to get serious desktop publishing done with accurate color, like InDesign, Photoshop and Illustrator. If you can't spend so much money and want an accurate CMYK workflow, take
Affinity Photo and
Affinity Designer and use them together with Scribus. If Scribus isn't yours, you could take a look at
Viva Designer or
Quark Express, but they are not cheap either.
Finally you'll come to the point to check your PDF for color, fonts and PDF/X compatibility. That's where Adobe Acrobat Pro is a useful app. There are still desktop licenses of Acrobat Pro DC, that would make sense if you don't have a Creative Cloud subscription anyways. Alternatively you could take a look at
PDF Studio, but I haven't tested it yet and can't ensure that it's a real alternative to Adobe Acrobat Pro.
To walk a step back, you could just work with the two Affinity products and forget about the DTP app, because for single page layouts like flyers Affinity Designer would be valid enough to get the vector art done, import pixel based images and export to CMYK PDF. But that's on you, because there is a reason for DTP tools, that can handle multi-page layouts and font setting much more comfortable.
Besides those apps you could probably take any other app that can bring your layout on screen and print it on your own laser or inkjet printer, but when it comes to 500 and up printouts, it's worth to do it as professional as possible and learn the basics of DTP or hire someone that does it all for you.