Making your own unique drinking glasses is a lofty ambition. It is quite an endeavor that requires at least some knowledge of glass blowing as well as access to the appropriate supplies and equipment. If you are not experienced with this craft and would like to create your own glassware you will need to search in your area for a school or studio that offers classes in hot glass.
Glass blowing is a bit labor intensive and not exactly a solo enterprise. There are many steps involved in hot glass and it is best to work with a team of other people. Creating your own drinking glasses will take some doing and you will have to put in a lot of effort to be able to create an actual set.
You could also recycle glass bottles to create drinking glasses by cutting off the tops and finishing off the rims. This will give you a nice assortment of tumblers you could then embellish via a variety of means. There are a few different techniques for cutting the bottles and you may have to experiment a bit to find the one that works best for you. There is an excellent book on the subject, "Bottle Cutting," by Michael Jean De Forrest.
In addition to exploring the bottle cutting techniques the book also shows how depending upon the shape of the bottle you can create interesting vases and other objects by cutting the bottle near the base and inverting its top and securing it in place with a special epoxy. While this is a great way to reuse and recycle it won't create dishwasher safe glassware and the end result will be very "homemade" in appearance which may not give you the look you want if you are going for something more polished.
If you are unable to find a glass blowing class offered near you and you would like something fancier than the recycled bottle tumbler described above then you may have to settle for getting inexpensive plain drinking glasses and reinventing them in your own image. There are quite a few options available for embellishing and personalizing existing drinking glasses including glass etching, engraving and adding color with special glass paints.
You may be lucky enough to find an inexpensive engraving tool at a dollar or discount store that you can use to create simple borders or monograms on your glassware. Be sure to wear a dust mask to avoid inhaling the glass particles and safety goggles to protect your eyes while working. To create your design on the glass you can use a sharpie or other permanent marker to draw directly onto the surface of the glass and then engrave your design. If you decide your drawing isn't quite what you want you can remove the sharpie impression with a little rubbing alcohol.
Etching creams are sold in hobby shops and craft stores as well as online. Companies such as Etchworld have entire lines of stencils and tools to create etched designs on glass using their acid based cream. Be sure to follow all of the safety precautions on the packaging. You can easily create your own stencil by drawing on self-adhesive contact paper and carefully cutting the shape(s) to be etched with a craft knife.
Adding color with one of the available craft paints for use on glass is probably the simplest way to embellish and personalize your own drinking glasses. Hand washing will more than likely be best but you should still probably choose a paint that is heat safe just in case. After selecting your paints choose your design and mask off the drinking glass with tape and paint.
Making your own drinking glasses will provide you with unique tableware that will make for great conversation at dinner parties and every time you look at your glasses they will give you a sense of pride in your accomplishment.