Why do I need to transplant?
Roots grow outward for stability, so they naturally collect at the edges of the container and leave much of the soil in the middle of the pot unused. This will cause slowed growth because the roots may restrict each other. Transplanting allows for a more efficient root mass, as the space in the middle of the final container will be used.
Allowing the roots to collect at the edges of your 1-gallon pot, will ensure a more efficient root mass when transplanting to a 3-gallon pot. Each time you transplant, you can also bury the plant up to the first node or branch to shorten it's overall height.
It might be worth mentioning that you can bury the plant low or high in the pot you are transplanting to as well; that way all your plants will start off being the same height. Transplanting high might be a good idea for patio growers in high rainfall conditions - this will help drainage.
Transplanting should be done a day after watering because moist roots and soil (not soaking wet or bone dry) will slide out of the old pot easier. The soil will hold together better, and less root damage will occur.
You should loosen the root bound roots from the bottom and lower sides of the root ball by teasing them out loosely and gently - this will help the transition. if there are too many hanging down after teasing, then it is ok to cut or tear a few handfuls of roots away. If the roots are wrapped around the root ball in the shape of the original pot, then it's best to cut these away to promote faster root penetration into the new pot.
The secret is the timing of the transplant; you want when the roots have filled the original pot, but before the roots have wrapped around and the plant has become root bound.
The danger of transplanting too early is that when you up can the pot and lift off the pot some of the soil and young roots may become damaged due to the fact that there is insufficient amount of roots to hold all the soil together.
The danger of waiting too long before transplanting is that growth will slow, as the plant has insufficient root room to match the above ground growth – ultimately yielding less. In addition, micro deficiencies, dehydration and other problems occur more frequently with root bound plants. Root-bound plants also take awhile to become vigorous again.
My preferred method of transplanting is to moisten the pots I'm going to transplant, moisten the new soil at the bottom of the new pots with a garden mister (the pump up models are best - misters are by far the best way to moisten the soil because they act like rain and do not compact the soil).
Mist each 3-inch layer of soil - lightly sprinkle around the old root ball in the new pot - this will create the best conditions for new growth. Roots will reach out and grow much faster into a light, airy moist soil than the compacted mud created by a hose or watering can to wet down the soil
I believe that many problems are caused initially by compacting the soil.
Lightly moisten your soil before and after transplanting. If a lot of water is pouring out of the holes in the bottoms of your pots when transplanting, then you are probably over wetting the soil and creating soil compaction.
If your soil is lacking in soil conditioners (e.g. sand, rock gravel, perlite, vermiculite etc) and the roots have penetrated the new soil - then water again. In normal conditions this would be 4 - 7 days after transplanting. Try to emulate the natural action of rain: pump up misters, watering wands, sprinkler type watering cans are best - forget about using a hose as this may compact the soil and it is also very easy to over water (quickly leaching out nutrients from the soil).
All the soil mix ingredients should be mixed in dry.... never stir or mix wet soil mixes, as they will turn to useless mud.