What’s the closest one can come to being able to do the things one does with Photoshop and Illustrator without paying $360/yr or a significant fraction thereof?
Do you mean besides GIMP?
Illustrator is for vector graphics. The best free version there is probably Inkscape.
No experience with that one, but GIMP covers >99% of what people do normally in Photoshop.
Depends on what you’re trying to do, honestly.
GIMP and Inkscape are both very capable, but GIMP can be tricky to use effectively and Inkscape doesn’t really give you the artistic controls of Illustrator. Since they are also both free, it is worth your time checking them out. If they do what you need, you’ve saved a bundle.
There’s lot of alternatives to Photoshop out there depending on your needs. Paint Shop Pro is one I know many folks swear by as a nice balance of power and usability.
Illustrator is trickier to replace. Corel Draw is the only one I know of that’s pro quality and still in development, but I think it’s substantially inferior in both capability and usability.
Photoshop actually has some nice vector tools in it for certain tasks. The cheapest way to get access to Photoshop is Adobe’s photography cloud program, which is something like $10-12 monthly and includes full access to Photoshop and Lightroom. Could be worth a look if Photoshop is the more important part of the equation for you.
Pixelmator is pretty good for $30.
Have you considered Photoshop Elements? Then again, you may not consider $60 “cheap”, and there’s no Illustrator equivalent to go with it.
Is CorelDRAW still being made?
I get the photo package from Adobe which is a subscription that’s $10 a month but gets me Photoshop and Lightroom both, and any updates that come along. I don’t own it but I like having the updates show up automatically. It’s a good price vs buying the programs outright.
It should be pointed out that this service (also referenced by typoink, is by subscription and, if I’m not mistaken, it is for one year ($119.40) at which time you can renew but Adobe has not set the renewal price yet. The current $9.95 per month is a special introductory offer for Photoshop CC 2014 and Lightroom 5. Both by the way are worth it if you have a lot of photos to catalog and edit.
Free online photo editor. Does a lot of what Photoshop does with nothing to download.
Fourthing GIMP. There is also the offshoot Gimpshop, which provides a lookalike Photoshop interface that would be more accessible to those accustomed to using Photoshop rather than learning GIMP’s comparatively clunky interface.
I’ve been using Inkscape for a little while to prepare graphics for presenting or publishing scientific data. I haven’t used Illustrator much so I can’t compare directly, however.
In my experience, Inkscape is a fine standalone tool for making figures, annotated images, and cartoon diagrams, but it has some quirks. It’s not very compatible with anything else I use – .pdf output is kinda screwy, and I haven’t figured out any good way to output vector graphics for use in any other program like Powerpoint or Illustrator. Right now I export figures as .jpg and then stick them into Powerpoint.
I use Photoshop and Illustrator (and InDesign) every day, and I can say that Photoshop Elements is REALLY solid. It’s not free, but for $60 or so, it gets you a huge percentage of the stuff that most people use Photoshop for.
Adobe Photoshop is one of those programs that is so powerful and has so many features that most people barely scratch the surface. PS Elements gives you all the basic stuff that people actually use and scraps a lot of the really high-end professional stuff that even a lot of us in the industry never learn to use.
If Photoshop is a Ferrari, Elements is a really nice Corvette. Until you can drive the Corvette to its absolute limits, you don’t really need the Ferrari. 
Corel PaintShop Pro combines the capabilities of both Photoshop and Illustrator in a single package, i.e., it does both raster and vector graphics.
EDIT: It costs about $100
Illustrator definitely can handle SVGs. I’m surprised Powerpoint can’t.
As for PDF output, if it’s screwy, then use a PDF printer instead.
This link has a great list of free alternatives: The best image viewing and editing tools - Gizmo's Best
Adobe claims it’s a permanent offer. It was an introductory offer for awhile, but this year they made it “permanent,” whatever that means. I can’t really believe they’ll offer it for $9.95 a month forever, but we’ll see. I’ve had it since November of last year (at that time, they said anyone who subscribes before XXX date will get the “permanent” deal. Plus, back then, you also had to own a licensed copy of CS5 or up to get the deal. Now, it’s available to first-time users.), so we’ll see, but I’m not expecting my monthly subscription to go up quite yet.
PNG format would be a better choice for that process.
There is no Mac version of CorelDraw, BTW.
From what I understand, Photoshop Elements does not allow you to covert from RGB to CMYK, which print shops often require. The place where my brother has his CD covers printed wants to charge him more than $100 to convert a color file to CMYK, so he has to send it to me so I can switch modes in Photoshop. I’ve never actually looked at the Photoshop Elements program, so I don’t know if my brother is just overlooking something, but my guess is that Adobe holding out the CMYK conversion is what requires professionals to buy the full version.
AFAIK the only vector image format supported by Powerpoint is WMF/EMF, Microsoft’s very own and very terrible format. And Inkscape’s EMF support is pretty dodgy, defeating the purpose of trying to keep things scalable and tweakable within Powerpoint.
Does Illustrator have better .emf support? Or is there a decent Powerpoint alternative with .svg support?
Inkscape makes OK .pdf files if you just need to print or share something, but not if you want to send it to someone else who wants to edit your files in Illustrator. In particular I’ve had issues with text formatting. Everything looks right if you convert texts to paths when you export a pdf from Inkscape, but then the recipient can’t edit the text.
[checks current temp folder full of images] Yup, png is what I’m using.