OK so I am late to this. But I have decent running experience even if I do it less these days. Maybe reading this thread will get me back out there. Not enough hours in a week... Or not enough strength in my legs for all the workouts. Who knows... Anyway.
I've run up to half marathon races, but I've done long fell runs (off road over all terrain) over 50km at a very solid pace. I've done 5k races and 10k races, never went under the 40 min barrier in a 10k race, 40:42 but managed it in training a few times. 5k somewhere in the 18 min range when at full speed, but never put much on that. Just did it for fun.
This is going to contradict a lot of the advice on here but... my experience:
If you are comfy in your shoes, dont change. Only change if something is wrong or you feel the heel deterioate. With conventional "running" shoes the heel is some sort of mesh or support and eventually will deteriorate. Likely you will get some foot pain on one side, just change the shoes out for the same ones and you will be back again.
This doesnt mean dont try other shoes, but dont expect the shoes to make you faster, they will improve your comfort and allow you to run perhaps more due to lack of any foot pain. Worth investigating but if you are in a comfy set, dont worry just yet.
The only way to get better at running, is to run. If you are out running you are improving. Could you improve faster, definitely, but just getting the shoes on and hitting the road will get you where you need to go eventually. Dont get caught up in feeling you missed your distance or your goal, come back the next session and try again. Running is a funny old game, some days you could be flying, next session you trudge round in minutes over your average pace. It dont matter, save the run on your chosen device, miles in the bank, get rest and try again. One bad run does not a bad runner make.
So training plans, I wont go there in depth but I recommend slowing down for sessions that are longer, speeding up for shorter, and reserving one a month where you try and break a PB over any distance. So race simulation. Interval training is also incredibly effective especially if you can do it over a set distance like on a track. 800m intervals are not only one of the most tough training routines you can do but they can really effectively increase your leg speed over 10k, they work for all distances but above 10k you need to learn to pace yourself and let the lungs take care of themselves.
You wanna run faster? You feel like you have hit your peak, that 50 minute 10k never goes lower than 49:30 no matter how fast you run... my answer was join a running club. Not a coach, a club.
Depending on the club size and dynamic will affect their setup (I am in the UK remember). My experience is a small club will have a set run, somewhere in the region of 10-15km with options to shorten or lengthen if you feel fresh. Then maybe a couple of paces, fast and medium with the medium being a more social affair not dropping anyone out the back door to trudge home on their own. If you get dropped from the fast group you join the medium. Simple. In the fast you have, let's say over 15km 7 checkpoints where the rapido people (sub 35min 10k guys) can either run back to the group or wait for the others. Then you basically blitz it as far as you can until you have to drop to the medium. This took me from a 44 min 10k to a 40 min 10k in less than 6 months. And it was so much fun racing people. I did one club run a week. But you could do more if your felt like it. I found one exhaustive run like that was enough, then I could do my own runs with less exhaustion (if there is someone to race, I race them. I could never not go in the fast group I always felt like a pussy, so to stop that I just didnt go and ran on my own, personality dependant).
Bigger running clubs will have pace runners in groups. They will have people put their hand up and say "I am running 10 min/mile", "I am running 7 min/mile". Loads, just pick one, and go. No waiting at checkpoints, likely no dropping people unless pre-agreed to be that way (you know the route and are regular runners). This was great for me too, I could test myself with the faster guys when I wanted, and go out for a paced run when I wanted.
Both club styles worked great, personally preferred the smaller one as I do enjoy a session of killing yourself trying to keep up with people you know you will never keep up with.
Whilst these clubs might seem like they tailor for faster guys that isnt necessarily true. Contact them up front and give them your 10k time or 5k time and they will advise if you can come along to certain sessions. You dont wanna turn up and be dropped in the first 500m. That sucks, been there... getting dropped at 3k, not too bad. A newbie to a club would normally have an escort of some type just to make sure you get round, or a non dropping run. Running (like cycling) in a group can be difficult to pace and catches you out for the first few, dont get put off.
I would say if you can run 5k, go to a club and ask. If you cant, just build up little by little until you can.
Quickly on shoes, despite saying you should just be comfy getting gait analysis can really help if you run a lot. You knees might be more grateful if you have the right shoes... Just dont expect a change of shoes to drop minutes off your time or to give you marathon distances. They simply aid your comfort and ability to recover/longevity in the sport.
Running mechanics in the body can take years to get right, it's all part of the journey. The more you run, the better your body will learn.