Cutting is commonly used method of reproducing roses as it’s easy and quick.

• The best time to take cuttings from plants is when they are growing vigorously, typically in early summer/spring to mid autumn.  Stems that are neither brand new nor fully mature and have fading flowers (or flowers that just lost their petals) are desirable. A stem with a flower bud showing no color istoo young.
• Select healthy wood: not too young or too old, no disease or pest- each cutting wants to be about 6" long, and the thickness of a pencil with 3-4 buds on it.

Pros : Taking cuttings is an easy way to quickly and cheaply increase your rose stocks.

Cons:  The plant will become weak and gives fewer flowers in 3-4 years

Some roses will do well on their own roots. Some won't such as white and yellow roses. But there seems to be a move by nurseries these days to grow roses on their own roots where possible. You will find that older style roses, climbers, ramblers and miniatures do well like this. Whereas Hybrid Teas can be too highly bred to perform well unless they are grafted onto a good rootstock.  

2. Budding/Grafing

Budding techniques can be considered to be grafting without leaves on the scion. When you take a scion with some leaves, you are doing grafting. Grafting techniques often require high humidity for the leaves not to wilt. This means the use of a misting system or plastic bag to keep the moisture.

• Mid summer is the best time to bud roses. You need the sap to be at full flow, giving the new bud a chance to "take".  Maybe a few weeks before mid summer is OK, as long as there is good strong fresh growth on the plants.
• The popular root roses are Rosa multiflora or R.indical (R. chinensis) because they’re tough and durable.

Pros : Fit for the weak rose to grow by the root of the wild strong rose.  Combing strength of 2 different roses: weak rose but having beautiful color or strong fragrance and stock rose which has strong root and can grow up well.

Cons: Budding/grafting is more complicate and you need to have the skill or experience to do it.   

 

3. Seeding

It’s not popular ways of reproducing roses because successful rate is less but at the same time, other ways are easier and faster
You can get the rose seed by collecting pollen from the stamens of one flower, and then use that to fertilize the stigmas of another.  Midsummer is the good time to pollinate the rose. With any luck, the hips will start to swell, and will be ready to harvest in early autumn (fall).  

 

You can find how to reproduce rose in detail in the below sources.  They explain the process and tips clearly.

Source: http://www.rosegardeningguru.com, http://www.love-of-roses.com