Sounds like you would be more interested in breadboard type projects where you put parts into a plastic module with lots of holes and conductive strips underneath, and the emphasis is on the functionality you are building, than with current kits in the heathkit tradition where the emphasis is on soldering it together rather than understanding and experimentally changing.
So you probably want to get a breadboard module and some assorted circuit components. There are probably assortments available right here, but basically you want an assortment of resistors, a dozen capacitors, some LEDs, at least a couple of NPN transistors (possibly PNP too), and ICs such as the 555 timer, perhaps some op-amps, a digital logic gate such as a 74LS00 or comparable, etc. And you will need some 22 or 24 guage wire and a power source, such as a USB phone charger, three very fresh alkaline AAA's, or a larger number of cells and a voltage regulator.
You can still find the basically discontinued radio shack spring-terminal electronics kits on the surplus market sometimes; those are good (and portable) within the constraints of the parts they have mounted in the kit, but it's harder to add additional components, especially ICs. With a breadboard you can just plug in anything with wire leads that can be persuaded to a multiple of 1/10th inch lead spacing, and if you run out of space you can stack a second breadboard beside the first.
You may also find the arduino boards a good way to extend to projects which include running software on a microcontroller chip. You could also start with one of the parts assortments that is meant for building add-on circuits to use with an arduino.