New cast iron skillet came out rusty after seasoning After looking them up for a long time, I finally decided to take the plunge and buy a cast iron skillet. 
After it arrived, I cleaned it in hot soapy water and scoured it, to remove the wax coating, as per the instructions. I wiped it dry, then popped it in the oven quickly to fully dry it. 
After this, I took sunflower oil, and coated the entire skillet. I then let it sit in the oven for an hour, at 200ºC. When I took it out, the coating was uneven (I now know I put too much oil), and was reddish-brown! 

I think I managed to get it rusty within the first day of owning it (pictures below)!

How do I resolve this, and ensure it does not happen again?
That looks a lot more like cooked oil than rust on the inside. Either way, probably not rust, and definitely not enough rust to cause concern. Scrub it off.

When I season my cast iron, I just do it on the stove. I heat the pan until it's smoking, and carefully add a little oil and swirl it around. Let it cool, wipe it out. Then, use it. When you do use it, clean it gently. Avoid using soap, just use water and a spatula or something. It's a good idea to heat it on the stove top to dry it well after cleaning. Over time you get a good thing going. 

But either way, rust with cast iron is no cause for concern. Even if it was completely covered in rust, you can brush it out and re-season.