Wanted to make sure I'm on the right track. For educational purposes, I wanted to measure the tau of charging up a simple LR circuit (inductor: 10.04 mH / resistor: 1013 ohm).

Using my oscilloscope, I put the first probe (Orange) on the positive side of the resistor and its ground on the negative side of the resistor.
The 2nd scope (violet) has the positive side of the resistor and its ground on ground (ground side of the inductor). See below.


The curve and charge time (tau and tau x 5) of probe 1 is what I expected (I captured a graph using a .999 mH inductor below but the numbers worked out close). The immediate jump of voltage (voltage leads current for a inductor) is what I expected (see below)


My questions: Is this the best way to measure current and inductor charging (I don't have a current probe) with an oscilloscope? Are there other methods (maybe the math function)?

 

Switch places, resistor & inductor. With the resistor on ground it will be better to measure voltage (current through the coil).
You need a current probe. But just moving the resistor will work.

 

Yes, interchange the resistor and inductor as ron suggested.

Note that the inductor will generate a spark across the contacts when the switch is opened, which can erode the contacts.
To avoid this, place a diode (e.g. 1N4148) across the inductor (cathode to positive side) to suppress the transient.