I did a lot of research to decide which photo printer to go for.
A4 or A3+. I decided that an A3+ printer would be ideal, especially as prices of A3 printers have gone down.
Dye or Pigment ? I understand that for Glossy paper dye is better, while for Matte, pigment is better. However in the real world I personally prefer the dye gradations to the contrasty pigment inks. I went for a dye + second pigment black. If you use CISS (continuous ink system) or re-fillable cartridges, you can obviously vary the type of ink, although I understand you have to then use some cleaning cartridges between type changes. I have never personally used an ink type (dye and pigment) that is not recommended by the manufacturer, and I do not know whether that voids the warranty.
How many inks. Most have CMYK (cyan, magenta, yellow and black). Some have two extra lighter cyan and magenta. Some have Red and blue, and others have different blacks or greys.
Droplet size makes a difference to both gradations of colour tones and resolution, and they vary between 1-5 picolitre, with most new printers between 1-3. Type of paper makes a big difference to result though, and one has to use recommended or original papers or paper from long-standing manufacturers using standardised products.
What resolution. Most photo inkjet printers nowadays have enough resolution to print very high resolution prints on up to A3+, and the resolution is also very much dependant on the droplet size and the print algorithm. I still lean towards the higher resolution machines for a more natural print.
Epson, HP or Canon. The three big players in photo printers. Epson use Peizo-electric heads, with a longer print cycles than the bubble or thermal heads used by HP and Canon. On the other hand, HP has head in cartridge, so every cartridge change includes a new head, and therefore more expensive. Canon has a very accessible and removable print head that can be easily changed, and costs about one third the price of the printer. Here the choice is difficult, although I personally tried all three.
Wi-Fi and network connection or not. This is now getting to the practicalities. I have 10 USB ports on my computer, and therefore went for the cheaper option of USB only, but for some, go for the network connection (Ethernet or Wi-Fi) if you need it.
Glossy or Matte paper. I prefer Matte for artistic effects. I also like textured paper and canvas. But for many users (portraits, weddings and the like) glossy is a better option.
Original or compatible cartridges, or CISS. For serious work I prefer original ink and paper to get the best results, although compatibles and CISS (continuous ink systems) are good for proofing and draft work.
Printing is a very serious business, if it is to be done properly, and requires a lot of calibration of equipment and standardisation of inks and paper, at least to get a consistent result.