^ Those two questions are really good points to start with.
I'll juse use myself as an example because it'll really be personal.
I started out with nothing except a DVD. It was Jillian Michael's 30 day shred. I didn't even have dumbbells when I started. I did the video, and then did it again. I went outside to run every other day back when I had the time for such things.
Now?.. My gym isn't THAT much bigger. I have 5's, 10's, and 8's in dumbbells, and some stretchy rubber things with a ball-stopper that you use to wedge into the door so you can have resistance, an exercise ball, a small weighted ball, 2 yoga blocks, a stretching-assistance rope, and a 20$ pull up bar for the door frame, a yoga mat, and a 15 lb kettlebell. I got all of these items for cheap/free from thrift shops and flea markets or friends.
I use Fitness blender and other youtube shows, a copy of every P90X series (still highly recommend those.. and I got these from friends so I didn't pay for them myself).. and that's about it really. I still run outside, but only in warm weather now.
My general goals are to stay in shape no matter what the weather is like outside (considering I have cold urticaria which makes exercising outside difficult most of the spring, winter, and fall here) and vague instances of "becoming stronger". As long as I feel like I am progressing or maintaining I feel alright with it.
If I were you? I'd start out with nothing, work your way up. There are a lot of things that require no weight at all, no equipment to get started.
A good set of dumbbells will get you where you need to go. I really like the fancy bow-flex style ones that you can adjust the weight on.. one day I'll splurge on them. you can go from 5lbs to 90 lbs pretty easily, and it all takes up one single space. But a good set of something that feels a hair too light (for when you burn out), and a hair too heavy will suffice. You can usually find adjustable sets with the circle weights on craigslist from people getting rid of their old stuff.
A yoga mat and block is pretty cheap, and would be my next suggestion. You can get away with a couple old towels for a while, but a good cushion makes ab work much better on your tailbone. You don't need to do yoga to benefit from these items, but stretching is very important and good form is easier to achieve with a block to assist and 'bring the ground up to you' so to speak.
After that, I'd get a pull up bar because there are sooo many exercises you can do with them. The one I have lets you swing it out and around, so you can do modified pull ups when you burn out (which is very very fast for most females so I like it).
Then you can start branching out with what you think will do for you.. for me, this is about all the equipment I really need. Maaaaybe a work out bench, just a plain old bench. The rest is fluff. Between all the exercises one can do without weights, with some light or heavy weights, and a pull up bar, the rest is just because I happened to have it or get it from someone. For you, you might want to start bench pressing weights, or squatting with heavier weights and you might want to consider an adjustable bench bar with some weights to stack onto it. Or you might realize you can just use a big old block of wood outside.
Whatever is going to keep you motivated and moving.. that's the essential equipment.