In February and March 2014 I participated in a Coursera course about the history of the Internet. One of the assignments was a 1000 word essay on what the Internet might be like in 20 years' time:

Write an essay that imagines how the Internet will be different 20 years from now. Justify your answer by connecting your ideas to the history of the Internet that we have learned in this class and through outside materials. Your answer can focus on how technology will change or how people will change or how governments and policy will change or even how society might change.

In 1000 words I'd have to leave lots out, so this page says roughly what I'd like to say. Apart from this page, I also have a page on signs of the times, occurrences confirming (or potentially contradicting) my predictions.

The Internet in 2034

To understand the future, look at the past

What is the Internet? I've already ranted about the fact that Microsoft seems to confuse it with the World-Wide Web. But there are various viewpoints. Clearly the essay topic implies an end-user viewpoint.

So what does the past of the Internet look like?

It came into being as an upgrade of ARPANET some time in 1983.
But it wasn't until NSFNET a couple of years later that people outside the military started using it to any extent.
In 1991 Tim Berners-Lee developed the first components of the web, but it took at least 3 years before it took off. I first saw it on 27 September 1993, and I found it boring.
In 1995 the last restrictions on commercial use of the Internet were lifted, and end users were allowed to use the Internet.

And basically, those were the technical and political milestones. The rest is evolution, not revolution. Round 1989 I started using the Internet as well. I had been living inside the Tandem corporate network for 7 years, and I felt comfortable with the environment. The big difference from Tandem was the lack of even the slight regulation that we had at Tandem, and the larger number of people on the network. But at the time the main use of the net was to move files around and for email and news. At Tandem we had globally visible file names that served the function of URLs, and we frequently made information visible on the net under specific file names.

The current status

The biggest change in the Internet was the introduction of graphics as a basic component. Until the advent of the World Wide Web, most Internet services were text-based. Hyperlinking had been around for some time, but it took the Web to make it useful. And since then, over 20 years ago, nothing fundamental has changed. People are still consolidating this basic functionality.

In more detail, since 1995—nearly 20 years—the biggest change has been the increase in network speed and capacity. In 1995, most users were connected by dial-up links that typically managed download speeds of 28.8 kb/s. Nowadays in Western countries, at least in the cities, speeds of 20 Mb/s and higher are generally available. In the process, there has not been a significant change in the cost of such a connection.

Most of the current Internet services are not new:

Commerce via the Web has been around for over 15 years. Amazon and eBay both started trading in 1995.
Social media sites were slower to take off. The oldest I know of is Orkut, which is now over 10 years old. It appears that it took some time for social media to gain critical mass. But that has now definitely happened.
Multimedia have been available via the Internet since before the web, but they have become increasingly important as network speeds increase. In most parts of the western world it is now possible to receive streaming TV, though the quality is frequently worse than broadcast.

Probably the biggest change since 1994 is the number of users: in 1994 almost nobody even knew about the Internet. Nowadays 77% of the population of the developed world uses the Internet. In the Scandinavian countries penetration is over 90%, in Australia, the USA and most of Western Europe it is well over 80%, as this page shows. In most countries broadband (here defined as a download speed of 256 kb/s or higher) penetration is less than 50%.

The future

In 2034, society will be centred around the Internet. People who don't use the Internet for significant parts of their daily life will be looked at like people today look at those who don't read newspapers or watch TV.

On a technical level, broadband penetration will increase. By 2034 the average Western household will have a connection with speeds of 100 Mb/s or greater. Speeds of up to 1 Gb/s will be available in some areas, but they will not be utilized to any significant degree: there's very little that an average household could do with such a speed. Other issues, such as network latency, will be more important.

These speeds are enough to allow a number of changes outside the immediate technical domain of the Internet:

Most purchases will occur on-line, and the few remaining shops will mainly exist to order and supply goods available on the Internet. The exceptions will be fresh goods such as food and some services where a view of the items in advance is desired, such as some clothes. This will also have a profound effect on the economy: many companies, notably shops and mall owners, will go bankrupt. The future will favour those who can adapt.
Radio and TV broadcasts will gradually cease. They are costly: they use a lot of power, and transmission and reception equipment are also expensive. Even now it is possible to receive TV via the Internet: a recent report found that in Western countries, 90% of Internet users watched online video. Significant investment in net backbones will be required to deliver the content, but it will still be cheaper than maintaining current transmission equipment. This change in the distribution method will also allow video on demand to take the place of fixed broadcasts.
Traditional publication will mainly cease. News sources such as newspapers and magazines are not as timely as the web, and distribution costs are an order of magnitude greater. One of the few technical innovations will be a way to display many pages alternately, like the reader of a newspaper or magazine can do by putting his finger in the corresponding place. Book publication may not cease completely, but it will be limited to high-quality publications and may be transformed beyond recognition.
The public switched telephone network will cease to exist. Even now voice over IP (VoIP), using the Internet for telephones has become mainstream. Services like Skype extend basic VoIP functionality to video telephony. By 2034 VoIP and similar protocols will be the only services available, and telephone companies will offer VoIP-based telephone lookalikes to those few subscribers who do not wish to use the Internet for other purposes.
Internet privacy will gradually become less powerful. People will get used to the fact that they are being “spied” upon, though of course many will take the consequences and not use services that they find intrusive.
Pornography will not go away. It will continue to be available to underage children. Parental controls will not work. Instead, social attitudes to the issue will change, just as they have changed in the last few decades regard to condoms.
Software and media licensing will continue to be a contentious issue. I would like to hope that predatory licensing attitudes of big corporations would be a thing of the past, but that's probably being too hopeful.

A couple of items are difficult to guess:

How important will social networks be? Will they reach a plateau or even decline? Or will they continue to grow and form a significant part of social life?
Will the ideal of telecommuting become a reality? With improved communications and higher transport costs, they could. But we've been saying this for decades already, and it hasn't happened to any great extent yet.