I've usually tried using "closeness" as a starting point. After all, topology education often begins with points of closure. If you plan on using a metric topology as an example, then you will have the intuition of a metric at your disposal for this.
That also sets you up for a good approximation for what continuity does: continuous functions preserve closeness (since they are preserving points of closure.) You can probably also riff off of that onto (path) connectedness, since a path is sort of like a string of points shoulder to shoulder.
Getting open sets across is a fun challenge. I've always found the picture of them being sets with "fuzzy edges" helpful. (It works best for metric spaces, but I suppose the picture is not universally applicable.) "Near" is a fuzzy notion after all, right? So if you are a point and you are "near" to another set by all measures of nearness that you have (these measures are open sets) then you are a point of closure to the set, and that makes you as close as possible.
I guess that means that you can try to convey closed set as having sharp, or well-defined edges. That helps a bit more with general connectedness, because "splitting" a space into two pieces with sharp edges makes it seem like the cut is a clean one. Clopen sets do not really play well into the fuzzy-sharp picture though :(
Compactness will probably always be a challenge. You can always try to motivate them as generalizations of finite sets. I know this can be problematic, though, since not only does it depend on the points of the space, it depends on how rich the topology is. After all, the antidiscrete topology on any space makes it compact, no matter how big the space is. I guess to get across this gap, you'd might fix a set and talk about different topologies on it. Specifically, you might use $\Bbb R$ with the order topology and then with the discrete topology. This gives you an opportunity to show what differences choice in open sets makes in the topology of a space.