The arbor is a generally simple structure. Two sides and an open-work roof connecting them, sometimes with a bench. Despite the simplicity of its design, arbors come in many variations. It can be metal or wood or vinyl, and there are many places to get arbors. Some places offer simple kits that can be delivered to your door. Making an arbor yourself can be a good project, but it requires a few basic carpentry skills.
The shading from an arbor comes mostly from the ivy or vines used to decorate it, or the plants hung from it. This lets you get creative, but gives you some structure to work within. Ivy and vines do require regular maintenance, however, or they could well take over your whole yard.
Consider also the option of a retractable canvas awning. The advantage here is the ability to pull the covering to block the rain, but retract it when you want some sunshine. A simple rolling canvas design doesn't require any heavy labor to create. You just need a basic wooden frame over your patio and then canvas panels attached to one side. The canvas can be rolled in and tied to the frame when it's sunny, and then unrolled and tied between the sides of the frame should it look like rain.
This could also be done between two trellis, or two arbors, to cover larger areas.
Trees and shrubs can provide both shading and natural beauty. When deciding on what plants to use, consider the plants that grow naturally in your area, as these will take best to the soil conditions and weather. Also consider the scents of the various plants and add an extra element of relaxation to your patio. The growing cycle of your trees and shrubs is also an important consideration. If you need the shade in summer, you don't want to use plants that won't be at their best just when you need them.
Plants are also more maintenance intensive than other options. Not only do they need to be watered and weeded, and trimmed, but there are other factors to consider such as falling leaves and bird droppings.
Pergolas are something like large arbors. They cover a much larger space and their ceilings can be solid as well as open-work or slatted. They come in a variety of material and styles, and can also be used for hanging plants or growing vines and ivies. However, pergolas also offer another option. Because they cover a larger space than an arbor you can hang canvass, curtains, or even blinds from their edges, allowing you to adjust the amount of sun you get, while still keeping you dry in the rain.
Pergolas require a more work to build than arbors, though, and they can be expensive in comparison to other options.