I know this probably isn't an opinion that many will aggree with, but personally, I think an oscilloscope is a better first tool than a multimeter (and you'll need to spend at least $400 for a scope). The reason I say this, is because for your blinking lights sort of stuff, knowing precicely what voltage you have isn't very important... a DSO will give you a ball park estimate of what it is (enough to know if it should light or not, be within tolerance for powering an IC etc) and you'll be able to see the frequency operation of your circuit.
The first two things I would buy would be a good 30V 3A power supply with current limiting and a digital readout, then a Rigol scope, a bench multimeter, and finally a waveform generator. A LCR meter also comes in handy. If you need more exotic tools later on like a spectrum analyzer or DC electronic load then you can worry about that when the time comes, but for now, I'd say give up on the $150 budget and getting a multimeter in general, and wait until your budget allows for you to buy decent equipment that will last you a lifetime. Your arduino and little LED projects are a great place to start and you generally won't need a multimeter for those things; and as long as you have a PC, programming can be learned for free.
As for what to spend $150 on... components!!! Lots of components. Get yourself some resistors, capacitors, inductors, op-amps, 555-timers, LED's and bins to store it all in.