Hello, I'm fairly new to running as well. Been going for 5 weeks only, with a week off for holiday, so a month or so of actual running.
When first starting, most people find it best tend to try to increase the time spent running rather than increasing speed, for building up endurance to do longer runs. Then you can graudually build up speed over time. There is a good program on this site that I've been following:
Week 1: Run 2 min, walk 1min, x7
Week 2: Run 5min, walk 2min, x4
Week 3: Run 8min, walk 2min, x3
Week 4: Run 12min, walk 1min, x3
Week 5: Run 15min, walk 1min, run 15min
Week 6: Run 30min
Seeing as you've already been running a few months already, you might not need to start at the very beginning, but it's a good guideline to get you up to running longer distances. I've found it very achievable.
I'm probably not the best person to ask, but I'm in a pair of slip-on trainers still (although they are very comfy!)
I would try to get decent shoes at some point, but personally I'm just making sure that this is something I will stick at long term before spending the money.
If you can, run outside, it's very different to on a treadmill from what I've experienced. I ran a little bit over the summer a couple of years ago outside, and when our school got a new gym the next year, we all had to try it out. When I got on the treadmill, it was MUCH easier than outside. You don't have to worry about pacing, slight inclines in the path and pavement sticking up to avoid...all in all less tiring. If you wait until you can yet up to 5k on a treadmill, then expect to be doing the same outside, it's likely it will seem a lot harder and could be demotivating, when in fact you're doing well for an outside run! It's a lot more interesting too, being able to choose where to go, explore new routes (without getting lost!) and see some interesting things on the way.
That's not to say treadmills don't have their place though. If it's icy, or a thunder storm, it's probably safer and easier to be inside on a nice treadmill! Also if you have a particular workout planned using the various inclines/speeds available it's a great use.
Hopefully that was of some help to you from a fellow newbie. 