I always seem to hit a wall in the game where I’m starting to expand out of Sol. Find a couple of worlds that seem decent and a bunch of mid systems that mostly are mining outposts.
There however never seems to be that much of a difference between my decent systems and mining systems.
How can I make my core worlds feel more like core worlds and not just larger mining outposts or naval bases?
It takes time and investment. You basically need to move larger amounts of population along with appropriate industries (factories, labs, shipyards, etc.). Typically, your “core worlds” in this sense will be your major production centers. It takes a lot longer to terraform a world and emplace a billion people on it than it does to plop 200 automines on an asteroid, though.
The game mechanics don’t really support high-population core worlds very much, since smaller populations are usually more efficient (higher fraction of pops in manufacturing instead of unproductive “service” jobs). I consider this unfortunate, since a mechanical incentive for high-population worlds would be good for roleplay in my opinion, but last time I asked Steve about it he said that he prefers the current mechanics to give players more reasons to expand across the galaxy (Aurora is, after all, a 4X game). Thus, I doubt this will change any time soon.
That said, having larger, centralized production hubs as core worlds does reduce micromanagement, and since suitable worlds for this purpose are relatively difficult to find there aren’t usually so many worlds to spread out across anyways, so high-population core worlds are likely to happen more or less organically given time.
Do you do anything different in actual space for your core systems or is it just planet side stuff?
The minimum efficiency of “workers/population” is achieved already at ~240M population, regardless of Colony Cost.
For example, if we take colony CC 0 (Earth), at >= ~240M population we will have 25% workers, and at 50M we will have 50% workers. If you want to create a really large production and keep a high percentage of workers, you will need to divide it into a very large number of colonies. For example, 200m workers is 800M population in one colony, or 400M in eight colonies (CC 0). You will also need to allocate resources between all eight. You will also not be able to allocate 50 laboratories to a talented scientist, since each small colony will have no more than 25 laboratories.
So I think you should consider this more as a bonus to colony growth than as a reason to divide your 5000 factories into 500 colonies.
In all honesty, my core systems are often less defined by population and industry - though my core systems almost always have both - than by my military and support infrastructure built. My core systems usually have the preponderance of my navy, army units, STO batteries, and the support to sustain all that.
This is more what I’m thinking about, I just feel like my systems need to have more going on in them.
I’m not very far into this play through yet but the only real difference between Sol and 4 settled systems is I have patrols in the outer belt and the number of ships sitting at earth.
I’m considering building much smaller cargo ships and have lots of them moving things around in systems. Maybe have asteroid colonies producing a small number of ship components I use a lot and having them shipped to earth.
All my major systems are the Sun and the captured capitals of NPRs. All the others are miners. 
One challenge that I often set for myself in games that have progressed quite far is to establish a secondary empire base somewhere far off. If you have maybe 20 billion km and 7 jumps between your capital and another colony, then you quickly develop a need for selfsufficiency, both for production and protection. Following the logic of what your colony needs at any one time quickly gives you natural goals to work towards.
I also try to specialize at least some colonies near the center, partly for RP, but there are also mechanical reasons for it. Fx Luna or Mars are often colonized quickly to trigger the civilian economy. Even if they don’t have minerals, I might colonize them anyway and find a different for them. Luna could become a production site for financial centers to secure specialized wealth production.
If Luna, Mars, Venus and Mercury have no minerals, I use them as defence front line, and install some STOs on them anyway: the ones on Luna, to support the ones on Earth; the ones on the other planets, to detract some enemy ships from Earth.