I have read that many hostels have kitchens or kitchenettes where guests can cook. This is something I plan to do. However, it's not too clear to me what's provided. Do I need to bring my own cookware?
It very much depends on the hostel. Some provide amazing kitchens with multiple ovens, fridges and every utensil you can imagine.
Others require a cash deposit to use pots, cutlery etc.
Generally, no, you don't need to bring if it's a highly rated hostel on hostelbookers or hostelworld - read the reviews and if there's a "great kitchen" comment, it's a good sign.
The lower end ones .. sometimes are lacking a bit in facilities. Some really rough ones haven't even had kitchens available for use.
The reviews are your friend - even if you don't book on the sites, it's handy to look through for comments about what you want- be it wifi, a kitchen, or warnings about bedbugs.
In addition to what Mark Mayo said, I would specifically call the hostel/extended stay hotel and inquire what quality the cookware is. Sadly most reviews dont dig this.
If need be I will specifically ask them if the cookware will stand me cooking X. I have had many bad experiences with a hotels/hostel providing cheap cookware which either burnt my food or overcooks. Basically throws your recipe out of gear and takes the joy out of cooking. The same goes for microwave related cooking. The wattage and the size and the quality of containers. This is just me, but I like to be specific so YMMV.
This varies completely from hostel to hostel. I've been to hostels in several countries, and there is no rule. Some hostels have kitchens with even ovens and dishes to cook, some have just some basic plates and cutlery, and some have no kitchens at all.
The only way to know for sure is to read if there is any comment on booking site or contact hostel directly.
When it comes to kitchen equipment, it varies. Many hostels, especially cheaper and offering paid meals don't have a kitchen or have only a kettle and a few cups. If the hostel site says there's a kitchen, you might find out there's only a microwave. They would then have plates, spoons etc., but any cookware would be useless without a stove.
I haven't met a hostel that would have a stove and no cookware, or a microwave and no plates. So you should expect that they would have the utensils matching the equipment they have. It would be a bit strange to provide a stove without any pots. But you should always try to call or mail the hostel directly in ask.
Taking cookware is risky. It takes a lot of space in your rucksack, and it might be useless, unless you take a water heater with your cup, or camping stove with your pot.