I would:
– put the tube inside the tyre before putting the tyre near the rim. Inflate it only very slightly so that it just holds its shape. This should prevent you pinching it on its way in; you can always let a little air out if it's tough to fit
– put the valve through the rim hole and fit one bead (one side of the tyre) starting at the valve and working round to the far side. You should not need a tyre lever for this bit. This side is now done.
– starting from opposite the valve, begin to fit the other bead. Work evenly on both sides towards the valve. When things begin to get difficult, go back to where you started on this side and push the bead further "into" the rim so that it goes into the well in the middle of the rim. This should give you a little more bead to work with near the valve side as the middle of the rim (in cross section) is a smaller diameter so you have a bit more bead available at the other side near the valve.
– be careful not to pinch the tube. If it's really tough to fit, let a little air out of the tube and see if it gets easier. You shouldn't need tyre levers.
Hope this helps. Like I said, that's what I would do. I've worked in the bike trade for quite a few years and have removed and fitted thousands of tyres, literally. Once you get into the habit of doing it "the easy way" then you can fit (and often remove) most tyres without having to use tyre levers – although, of course, a few tyres are really awkward even _with_ levers.
One last thing: when you inflate the tyre, put only about 15-20psi in it then check all the way round the bead on both sides of the tyre to make sure it's seated properly. The ridged or striped band on the tyre should be evenly visible all the way round – any narrow bits or bulges and you need to pull the tyre about to even it out, otherwise you increase the risk of a blowout or tyre rolloff.
: P