Hybridizing:  Plant propagation has two general types: sexual, and asexual or vegetative.  Sexual propagation results in seed.  It involves two parents and the seedling is a genetic blend of the two parents.  Sexual propagation is usually done when the favorable characteristics of two parents are wanted on one plant.  This is the field of "hybridizing".  It sometimes results in markedly improved plants, but a great deal of trial and error is involved.  Once a successful hybrid has been created, it must be propagated by asexual methods to make genetically identical plants.  Asexual or vegetative propagation is when vegetative tissue is nurtured into a new plant that is genetically identical to the original.  Of course, a graft has different roots and top, but the top is genetically identical to the scion.  However, the rootstock may cause variation from the original plant, such as dwarfing, disease resistant and tolerance of near alkaline conditions.

Seeding:  Gather seed capsules in the fall when they turn brown.  Allow them to dry, remove and clean the seeds, and keep them in an envelope.  In February, sow the seeds in a small pot containing 50% milled sphagnum moss and 50% horticultural perlite.  Do not cover the seeds with the medium; just sprinkle the seeds on top.  The germinating mixture should be sterilized with boiling water and allowed to cool before sowing.  The pot needs to be in a controlled humid environment.  Polyethylene bags are great for maintaining a high humidity.  The pot is placed in a polyethylene bag with stakes to keep bag away from the germinating seed and placed in a light area with no direct sunlight.  The pot is rotated once or twice a week to compensate for variations in light and temperature.  Generally, no winter chilling is necessary to get the seeds to break dormancy. Keep the seed pots or flats in the 65°F to 75°F range. The seed will sprout in from 3 to 8 weeks (sometimes longer), and after they have formed their first set of true leaves they can be transplanted into other flats or containers.  It's important to open the plastic bag gradually.  Since they are so tiny, they will not stand frost until they are larger and some months older.  The seedlings will need to stay in the flats for probably 2 years, after which they can be transplanted into open ground.

Cloning:  Propagation of hybrid plants and selected cultivars of species requires a form of cloning.  These include cuttings, tissue culture, grafting, and layering. Most rhododendron and evergreen azalea cuttings root fairly easily.  Deciduous azaleas require special techniques to root.

Cuttings:

1) Most rhododendrons and evergreen azaleas may be propagated from stem cuttings.  Cuttings are usually taken in the early fall from new growth that is just beginning to harden off.  Generally, softer wood roots more readily than harder wood, though the softer the wood, the more likely it is that problems will occur with fungus-related diseases.  Cuttings are taken in the morning when full of moisture.  The cuttings are usually terminal cuttings with one whirl of leaves and the leaves cut in half (to reduce the leaf area), and any flower buds removed.  Wound the cutting with a cut on each side, about 1/2" to 1" long, just deep enough to remove a sliver of green bark.  Dipping the cutting in a rooting hormone containing indolebutyric acid will aid rooting.  The cutting has the end cut off just before dipping in rooting hormone containing a fungicide.  Then the cuttings are placed in a flat of sterile media containing a mix of 50% milled sphagnum moss, and 50% horticultural perlite or vermiculite.  The flat is placed in a polyethylene bag with struts to keep the bag away from the foliage and placed in a light area with no direct sunlight.  The flat is rotated once or twice a week to compensate for variations in light and temperature.  Using bottom warmth of 70-75°F will encourage root growth. Rooting usually takes about 6 weeks for evergreen azaleas and 3 to 4 months for large-leaf rhododendrons.  Once the cuttings have rooted, pot or transplant them to flats containing a sterile mix of 60% milled sphagnum peat moss and 40% perlite.  Fertilize once a month with an acid-based azalea plant food like Peters.  Removing terminal buds promotes sturdy well-branched plants.

2) Deciduous azaleas are very tricky to propagate from cuttings.  Take cuttings of deciduous azaleas when the new growth is soft and pliant.  This is often coincident with time of bloom in early June. The ability to root decreases rapidly as new growth matures.  Trim cuttings below a node (overall length of cuttings 3 to 5 inches) and dip in a rooting hormone containing fungicide.  Insert in a medium of 60% milled sphagnum peat moss and 40% horticultural perlite.  Usually bottom warmth of 75°F is used to encourage root growth.  In late August, transplant cuttings that are rooted and grown on in the greenhouse with supplementary light (14-hours a day) to prevent dormancy and induce new growth.  In the following fall, transfer to a cool, frost-free (35°F to 41°F) environment to induce dormancy.  As new growth develops in the spring, transfer plants to a shaded environment.

Layering:  This is the easiest form of propagation for the home gardener.  A lower branch is held down on soil (not mulch) with a stone.  A slit is cut in an area in contact with the soil and the cut is treated with a rooting hormone.  Then the cut area will sprout roots.  When the roots are developed enough to support the end of the branch, usually in 2 years, the rooted-branch is cut from the parent plant and transplanted.

Grafting:  Grafting was the standard propagation technique prior to the 1950s, and is still popular in Europe where acidic soil is not common and plants that are hard to grow can be grafted onto rootstocks of plants that are easy to grow like R. 'Cunningham's White'.  The big advantage of grafting is the use of disease resistant, soil tolerant rootstocks.  Also, some hybrids don't produce good root systems, so if the plants are grafted onto a good rootstock, this is no longer a problem.  There are two basic techniques of grafting, one involves grafting onto rooted rootstocks and the second involves grafting and rooting the rootstock at the same time.

1) With a conventional side graft, the rootstock is rooted and grown for about 2 years before used in the graft.  Then the scion is grafted to the rootstock and secured with an elastic band. See the referenced articles below for detailed instructions.

2) The simultaneous graft and root method involves rootstock that has not been rooted.  The cuttings for the rootstock and the scion are taken at the same time.  The scion is side grafted to the rootstock cutting and secured with raw cotton string which rots naturally.  The top is left on both cuttings. The rootstock is rooted immediately after the graft is secured.  After the rootstock has rooted, its top is removed immediately above the graft.  It doesn't matter if the scion forms roots but the rootstock must not form shoots.  Any such shoots are removed.

Tissue Culture: Tissue culture or micro propagation is a popular method of producing large numbers of rhododendrons for commercial production.  Over simplified, it involves taking a small vegetative shoot section from the parent plant and putting it into a test tube.  Through the use of agars and auxins, absolute sanitation, proper temperature and lights, the vegetative shoot is induced to grow into multiple "seedling-like" growths with no roots. This tiny juvenile vegetative shoot is then rooted.  Needless to say, this method is not for the general home gardener, though some "kitchen tissue culture" is being done (see reference article).

For further information on propagation and hybridizing rhododendrons and azaleas consult the following resources:

Propagation
  How to Grow Rhododendrons from Seed by Allan and Shirley Anderson
  A Modern Method for Propagating Rhododendrons From Seed by Peter G. Jordan
  Propagating Native Azaleas by Earl A. Sommerville
  Notes on Growing Media for Rhododendrons by J. L. Rouse
  Tips For Beginners: Success With Rooted Cuttings by Frank Dorsey
  Rhododendron Propagation from Cuttings, A Review by Warren Berg
  Grafting: A Surgical Cure, Part II: How To Do It by Bob Bondira
  Rhododendron Cutting-Grafting by Dr. Henri Galibert
  An Amateur in Vitro: Tissue Culture at Home by Donald W. Paden

Hybridization
  Pollinatimg Rhododendrons with Joe Bruso  YouTube Video
  Mechanics of Basic Hybridizing by Albert J. Muller
  Tips for Beginners: Hybridizing Notes, Part I, Part II by Jim Barlup