"Cheap" ink is a relative term. You have to consider the economic model of the printer business: The low price of the printer is subsidized by the high price of the ink/toner (like getting a free razor and paying a high price for the blades). That's how everyone in the business does it - you won't find a manufacturer who sells printers at a higher price and then sells ink at a lower cost - they can't compete feature-for-feature with the lower price of the ink-subsidized printers. Smaller/cheaper printers will also have cartridges with smaller capacity - some money can be saved by buying higher-priced printers that use higher-capacity cartridges, but if you don't print all that much, then you may not save much in the long run.
For better photo quality, you may end up with a printer that uses more than four colors of ink. A basic color inkjet printer needs 4 colors - black, cyan, magenta, and yellow. Printer makers will add additional "photo" inks to enhance quality. I currently have an Epson XP-830, I think it was around $129 - it has 5 cartridges, and does AirPrint. I'm not in love with the thing - it has a way of disconnecting itself from the WiFi network, its user interface is from the Dark Ages (despite the LCD touch screen), and the mechanics are fragile (my aging father managed to break off one of the ink cartridge release buttons) - but it does make very nice photo prints for the price.