BNC are very common in the electronics lab, but understand that not all things with BNC on them are necessarily inter-connectable.
Typcially, a BNC connector and the cables have 50 Ohm characteristic impedance. Likewise, the inputs and outputs of the gear with BNCs will have 50 Ohm impedances. This impedance matching is necessary for signal fidelity. It avoids reflections, resonances, and other nasties.
Most oscilloscopes are an exception. Although they use a BNC connector, they will have a high impedance input (1 Mohm in parallel with some few pF capacitance), and typical passive scope probes are designed to drive this impedance. There is in general no harm in connecting a 50 ohm source using a 50 Ohm cable to an oscilloscope input, but it will not be terminated (into 50 Ohms) by the oscilloscope. This means that the signal will be double the expected amplitude, and it will reflect back to the source. Most sources can handle this, but there can be exceptions. Some oscilloscopes have a switchable 50 Ohm termination, and you would usually choose to use it in this case, but must disable it when using passive probes which need the 1 MOhm input impedance. If the scope doesn't have this, then you can use either a feed-through terminator, or a BNC Tee with a terminator connected to one side.
Banana leads are very handy to have, I'd recommend getting a set of various lengths and colors. Retractable shrouds are perfectly fine, compatible with both normal and safety type jacks. They can help prevent "oops" moments.
You can get Banana to BNC adapters, and it doesn't hurt to have a couple around. Keep in mind that once you have a signal on flying wires, it is more susceptible to noise and won't work well at high frequencies (>> 1 MHz).
Note that some BNC cables have 75 ohm impedance. These may be identifiable by a distinctly different looking BNC connector which has no which plastic dielectric surrounding the center pin, but also they may not, as many are built using 50 ohm connectors for some strange reason. Always look at the cable itself to determine impedance. Or measure it. You mentioned having at least one pulse generator, this can be used with an oscilloscope to make a simple TDR measurement. You also need a good BNC 50 ohm cable, a BNC Tee, and a homemade variable terminator. I built one of these from a small 100 ohm pot wired as a rheostat to a BNC plug (or jack). It's amazing how quickly you can learn about the relationship between signal integrity and impedance when playing with such a rig. I was also able to measure the impedance of a few mystery BNC leads I had acquired. It turns out they are 63 Ohm. Who would have thought?