What you are describing may involve the clutch, the gearbox, and the throwout bearing. If the gears clash when shifting while stopped with the engine running, that sounds like clutch adjustment or a problem with the throwout bearing. At any rate, mashing down on the clutch is not completely disengaging the transmission. Double clutching is not applicable here. Adjustment problems of the shifter mechanism and wear in the gearbox is less likely, but possible.
Shutting down the engine is not really a solution to clashing gears during shifting, because, with the transmission in the gear you plan to start out in, even with the clutch mashed down, there is a lot of mechanical drag while you are running the starter. The recommended procedure for starting is to set the emergency brake, place the transmission in neutral, then start the engine with the clutch pedal released. This reduces the mechanical drag on the engine during starting with less load on the electrical system/starter. So, you got a "catch 22" connundrum.
But I digress. After you get moving in first (or second) gear, double clutching enters the picture. It is a learned art.
Once you accelerate to a speed and engine speed that is right for shifting to a higher gear, mash down on the clutch, take you foot off the gas, and move the shifter lever from the gear you are in to neutral. Let the clutch ALL THE WAY OUT and mash it right back down. Now finish shifting into the higher gear. Sometimes it helps while you are in neutral to mash and release the gas just a little bit to increase engine rpm, then let the rpm's fall off while finishing the shifting.
Downshifting, say from 4th to 3rd, or 3rd to 2nd gear takes a little more practice, but the the double-clutching procedure is the same, just a little more emphasis on matching engine rpm to gear- and drive-train speed when moving the shifter lever from neutral into the lower gear.
If you get really brave, you can up-shift (from 2nd to 3rd, for instance) with the double-clutching without using the clutch! Don't try it going up hill! And don't try to downshift from second to first while moving.
Best advice is to fix the problem (clutch and throwout bearing) rather than circumvent the solution by dancing with symptoms.
