Few people actually print their pictures anymore. Personally I find that a sad state of affairs. A couple years ago I was sharing a new project of mine with one of the top portrait photographers in the game and he asked if I had printed them yet. I told him I hadn't and he reprimanded me pretty swiftly. He said, "Print out a master set. Sign them, date them, put them in a box and put it on your shelf. Those are now the project." And he was right, you know. I came home and over the next couple of days I made an 11x14" print of each of the images and they sit in a box on my shelf as we speak. They're a physical product of the hundreds of hours of work I put into them. They're the end result.
Printing at home can be a lot of fun too, but there are numerous decisions along the line which can effect the final product. For the sake of this essay I'm assuming you've got yourself access to a nice pigment ink printer so that your prints will have a longevity measured in decades or centuries as opposed to weeks and months. I'm currently using an Epson R3000. It goes up to 13" wide which is plenty for me and uses larger ink cartridges than the R2880.
The biggest question is what paper you're going to use. Here are my thoughts.
Glossy Glossy paper is very modern and in many ways best represents the true photograph. It's got deep blacks ( and fully saturated colors. The problem is that it's quite reflective and can sometimes be like looking at an iPad on the beach. I personally don't use it much. Too shiny for my taste and somehow doesn't have any character. It's too clean.
Semi-Gloss Also known as satin, pearl or luster; semi-gloss papers are coated like glossy papers are, but the surface isn't smooth. Instead then typically have a textured, almost sandblasted finish which reduces the biggest problems of glossy. Imagine your average textured photo print from the 1970's. That probably had a satin finish. This is my preferred paper, not only does it have a D-max almost as high as glossy so that blacks and colors pop, the finish also drastically lessens the reflective problem. Best of both worlds.
Matte Matte papers are uncoated with a smooth finish. Think of them as a think piece of regular paper. A lot of people including my friend Jeffery love matte papers when printing black&white pictures. He says is reminds him of uncoated fiber papers from his darkroom days. I can see that, but when I print the same image on matte and satin and put the two next to each other, I almost always choose the satin. Blacks and color saturation on the uncoated papers are just to subdued to me. It's like someone turned down the vibrance and pulled up the black level to 25.
'Fine-Art' I'll admit, I don't get so-called 'fine-art' papers. Typically they're a more more rough, made from cotton rag, and often slightly tinted matte paper with more natural textures. I don't like the idea that my paper is purposefully changing the image I'm printing on it. If I wanted texture I'll do it in photoshop. Perhaps the argument is that buyers look at 'fine-art' paper and it looks more hand made or unique and therefore worth more money. I'm not sure, but it always looks hokey to me. Like it's trying too hard.
A paper's weight typically denotes it's thickness and in America is expressed in pounds. For example 68lb stock. Interestingly enough, this is derived from the weight of a ream (500 sheets) of uncut stock used during the manufacturing process.
How thick you like your paper is subjective, but I tend towards the middle. Too light and they're easy to bend and crease, too tight and they're hard to roll for shipment and flatten out afterwords. Medium thickness also works well in portfolio pages or stacks of prints for people to flip through.
Over the years, people have looked for ways to make paper a brighter color of white. So 'white' that they often have a blue tint to them. They do this by adding optical brighteners to the papers. It's the same stuff they put in laundry detergents to make your clothes colors pop. For longevity sake you should try to use the less brightened paper as a general rule of thumb.
You should also look for acid-free papers if you're anal about how long your images will be around. This usually limits you to matte and fine-art stock. A good matte paper and pigment inks and your prints will out last you by a long margin.
Personally, 90% of the time I print with Red River Ultra Pro Satin. It's got a great surface and D-max without the too many brighteners which make my eyes hurt. This is what I use to print images I sell. When I do print matte I go for their Polar Matte.
In the spirit of full disclosure, Red River Paper is a sponsor of the On Taking Pictures podcast but I've been using their products for years before any of that happened. They sell great paper and great prices. In fact, go to http://redriverpaper.com/otp and you'll get a great deal on a sample pack and 10% off when you decide to order.
Now go take some images and make some prints that you can pass down to your children's children.