Since you already have some experience in the uni lab, you should have an idea what you need. My $0.02 cents, YMMV depending on your projects.
Pomona has good stuff. Other good brands are Hirschmann and Multi-Contact (both are European, so may not be available in your part of the world).
Obviously one scope probe for each scope channel, 10x is fine, I rarely use 1x, it's only for signals that are both low frequency and low level (eg. power supply ripple), plus optionally one for the external trigger input if your scope has one. Also some banana to probe DMM leads (you probably got them with your multimeter). Both BNC-BNC (all 50ohm unless you're doing video) and banana-banana jumper cables (length depends on your workshop setup), you need several for things like series setup (eg. power supply -> DMM to measure current -> DUT -> power supply).
You also want to be able to clamp your DMM/power supply leads onto something, especially if you're using multiple DMMs at the same time (unless you're an octopus). Alligator clips are OK for large stuff, I prefer grabber hooks and tweezers. Alligator clips are useless for even something large like a TO-220 package or a PCB-mounted 1/4W resistor. Keep in mind that everything is going SMT, so being able to clip on say SOIC is very useful (same for even smaller stuff, but things get pricey fast). I prefer a modular system (eg. clips with banana jack connectors) to leads with integrated clips, but both have their uses (the connectors makes them heavier). You also want to be able to clip the output from your function generator to your DUT, you can abuse a 1x scope probe for this if your load is high-impedance, or get a BNC-to-banana jacks, BNC-to-grabber hooks or BNC-to-alligator clips cable.
You also want adapters between any type of connector that you use (eg. BNC to banana). Plus miscellaneous BNC hardware like terminators and tees.