You have lots of possibilities. (But there shouldn't be anything to "burn-in" and very likely there shouldn't be anything to burn out.)
Like Lightages said you can start by doing the probe compensation; that's a very good place to start.
You can use your DMMCheck Plus with the scope. Connect the probe to the V+ and the probe ground to the V-. You will be able to read the 5V (I think Vrms) on your scope when in the AC mode. In the DC mode will see 5V DC on the scope. I think you have to turn the DMMCheck on and off in order to switch between AC and DC.
You can also check frequencies with the DMMCheck. When you power up in the AC mode I think you will see 60 Hz. If you go back to DC you will get another frequency (not sure what yours is set for, maybe 20 kHz?)
You could do all of the above on all 4 channels. It will help you learn the some of the basic controls (Vertical, Horizontal, Trigger, etc.)
You might find that the DMMCheck and your scope are roughly in agreement but maybe not exactly. I wouldn't stress over it too much - you can do that later but first you should get comfortable with the fundamental operations of the scope to make sure you have eliminated user error before worrying about your gear or the last decimal place.
After you get the DMMCheck run through it's paces you can move on to your Wavetek.
Use the function out - it is 50 ohms. Be sure your Rigol is set for 50 ohms (I don't know if the 1054Z is automatically set or can be manually set to 50 ohms; on older DS2072 models the scope only had 1 Mohm inputs so you would need a 50 ohm feed through terminator). Make sure you know if you are using 50 ohms or 1 Mohm - it will influence your measurements. Assuming you are set for 50 ohms the Wavetek and Rigol should be in "agreement", but nothing bad will happen even if you are set at 1 Mohm on the scope. You can use that nice big dial on the Wavetek to change frequencies from 0 to 20 MHz. On DC you will get a horizontal line. With the attenuation knob I think you will see your line go up and down as you change the voltage. (I'm just looking at a picture of the 145, I haven't used one; you might have to learn to convert between dB settings on the sig gen and voltage measurements on the scope). Next, you can move to the sine wave setting. As you change frequencies with the big dial you can then adjust the horizontal Time Div knob on the scope to see sine waves narrow and widen. You should begin to get comfortable with how the graticule markings equate to the signals generated by the Wavetek. Work on your ability to trigger using the scope controls. It looks like you also have trigger controls on the Wavetek; I'd be inclined to learn to use those after you learn to use the scope trigger. If you twiddle too many dials all at once it might become a head scratching multi-variable problem to solve - which is ok if you like to make things more complex or if you just like twiddling dials.
After a while it should all work fine - in fact it will probably work fine from the get-go, but there is a learning curve which will help instill confidence. With just the Wavetek you should be able to keep yourself occupied for days if not weeks (or possibly longer) learning to use all the controls. Try to anticipate what is going to happen and then see if that happens as you twiddle the knobs. When you get sine waves figured out try square waves and then pulses. I think you are going to have a bunch of fun.
Read the manuals and keep them handy for consultation; if you don't have them try downloading them on the web.
Before everyone piles on here with their recommendations on what to acquire next I'd consider a breadboard, wire, resistors, LEDs, and a small power supply and a basic DMM - and then work to dial-in Ohm's Law. This should keep you busy for weeks to months; and there is enough remaining to occupy an entire life time
Enjoy!