I'm a huge fan of AirBnB! I use it ALL the time. I like that you can rent anything from a tiny room in someone's home to having an entire place to yourself. There are a ton of search options available, you can broaden your search to an entire city or narrow it down to a specific neighbourhood. It's great because you either get to stay with a local host, or in a cute neighborhood, or near a specific attraction you want to see. I tend to gravitate towards AirBnB so that I can get to know local people - plus they often have great tips for what to see and where to eat!
You should always read the reviews, AirBnB is based on reputation - if there's a ton of great reviews and one or two bad ones, it's probably that the one or two are just being overly picky. But, if you find an awesome place that doesn't sound like it's quite legit, and the reviews don't add up, it's probably not an ideal place (for example - we were looking to rent an apartment for the weekend in southern Alberta, and this gorgeous place came up in our search for super cheap, but the *only* review the place had suggested that the host was super sketchy and that the place was not as described).
Because AirBnB is based on individual people's homes, and we are all human, it's always a good idea to have a backup place or two just in case - *very* occasionally you will find the perfect place and try to book it, only to realize that the host has actually forgotten to update their calendar and it's not actually available. I like the wishlist function which lets you save multiple places based on specific dates and locations. Also keep in mind that depending on the area and date selected, there may be a lack of available options (for example, if you tried to book a place last minute in Pasadena during the RoseBowl you may find there's nothing available, or in many areas of Iceland they are few and far between).
My go-to hotel search site is Booking.com. I like their selection and their search options: you can search by list or using the map view, plus there are all kinds of filtering options available to you (budget, hotel rating, fun things to do, free cancellations, facilities, etc).
I have not stayed in an actual hostel, however, I have stayed in a few hostel-type hotels. Hosteling is definitely the way to go if you are on a budget and solo! Keep in mind that if there's more than one of you traveling it can actually be more expensive to stay at a hostel (because they generally charge per person instead of per room). I highly recommend hostels for solo travelers and younger travelers not just to keep your cost down but also as a great way to meet new people! You just might find a new friend to come check out the sites or grab a bite to eat!
If you're on a budget, solo, and looking to stay with locals, then couchsurfing is a great option! And the best part? IT'S FREE! Couchsurfing is an awesome way to meet locals who have opened up their home to provide a crash pad for a couple of days to a few weeks depending on how generous the host is. You will likely need to contact multiple listings before you find one that works with your travel dates, they recommend contacting around 5 potential hosts - just in case your top pick doesn't pan out, then you have a backup (or 2 or 4 lol).
Housesitting, petsitting, or houseswapping/home exchange is a great way to see the world without having to pay for one of the most expensive aspects of traveling: accommodations. If this is something that interests you, I would suggest starting locally and build up some references. I regularly housesit and dogsit for friends and family, and have branched out locally to improve my reference base.
Here is a great comparison from halftheclothes.com of some of the best and worst housesitting sites around the world.
HomeAway is similar to AirBnB but not quite the same. Individual owners list their homes on HomeAway (and VRBO and VacationRentals.com - all three are owned by the same company, but have different listings available on each unless the owner has cross-posted between them). Both HomeAway and AirBnb have hundreds of thousands of listings to choose from around the world, and both have excellent review systems in place so you know what you're getting into. The main difference I've noticed between HomeAway and AirBnB is that there seems to be more pet-friendly listings on HomeAway. HomeAway also tends to be more oriented towards entire homes or apartments, whereas AirBnB you can rent anything from a couch to a room to a house.
Accommodations are a very personal thing, it depends what you want to get out of your trip and who you are traveling with as to which type of accommodation will best suit your needs. An individual may prefer something like AirBnB or a Hostel or Couchsurfing, where you get to stay either with locals or meet other travelers during your stay. A family or a group may want more space and privacy, so renting an entire home through something like HomeStay, VRBO, or AirBnB might be the best option there. Hotels are great if you are only going to be there for a night or two and are looking for flexibility with check-in times and such, but I personally don't like staying in a hotel for extended periods of time because it can be hard to get a feel for a place that way.
Traveling with pets throws a whole new curveball into the mix as many places are not pet friendly. They can be encountered though, I have found that many Best Western Hotels are pet friendly, and some AirBnB's, and many of the HomeStay's/VRBO's/VacationRentals.com are also pet friendly.