How do I promote myself, as a Photographer

I suggest you get hold of the late Rohn Engh's book, Sell and Resell Your Photos. It has been the bible for freelancers for years. Lots of info in it on promoting yourself. Personally, I don't think social media is effective...and I have nearly 3000 Facebook friends.

As for cameras, consider other brands than Nikon and Canon, who have both lost their edge in my opinion. I've been a pro for over 40 years and use a couple of Sonys and just ordered an Olympus OM-D E-M10 Mark II Mirrorless Micro Four Thirds Digital. The E-M5 would be a good choice for a less expensive model. Olympus lenses are known for their sharpness and they are very compact, which is why I'm going with one over a light, but not as light, Sony DSLR.

Some great answers here already!

As a photographer - the most important thing, of course, is showcasing your top photographs. There are MANY ways to do this and here's what I would suggest:
Create a Facebook Business Page
Create an Instagram account just for your business
Build a portfolio website
Utilize a service, such as flickr, to load your photos
Get a great business card - hand these out at networking events, to people in general.

Upload photos by "type" of photography into Facebook
Highlight your best photos on Instagram

Based on the type of photography you enjoy, too, there are many niche services - do some research and get involved with ones that speak to you. It may even make sense to create a profile on Smartshoot. :)

The best thing you can do is be active on social media. Promote yourself through a Facebook page and invite all of your Facebook friends to "Like" your page. Send out weekly updates and showcase your work regularly. Instagram is a great way to push some photos out there and always look at hashtags to see people's best work. Depending on what you like to photograph, get in the know of what's trending. Camera bodies aren't necessarily has important as your glassware, but you need to know what you want to shoot. A nice cheap lens is the 50mm 1.8 and you can use it for a lot of different bodies. I use Nikon which is great and the decision is up to you. Typically, the equivalent Nikon Body is cheaper than that of the Canon's, but the Canon Lenses are normally cheaper. The best thing you can do is just research what you think would work well for what you want to shoot.

Let's start with the basics. Do you have a portfolio of your very best images? What do you like to shoot? Do you have a website? Are your images professional? Do you own a camera? The reason I ask these questions is to establish you have something to advertise. You need images to show you can produce good images in whatever category you choose to seek work. Once you have established what type of images you want to produce, then you can start marketing your work to the people who have a need for your images.
Regarding cameras, I'd start out with a used DSLR with a zoom lens. At this point, brand is not too important. Maybe there is a camera store in your area that sells used gear.