As you surmise, the problem is in cleaning/purifying the water we want to use. Desalination is extremely expensive, to the point where it's cheaper to have a war rather than build a desalination plant. This is currently most obvious in the middle east but changes to the Indian Ocean monsoon could have dramatic effects all around that area. India is working to mitigate the likely effects.
Desalinating Seawater is expensive. To give you some idea of numbers, they built a desalination plant in Victoria (Australia) recently, and that cost about $5 billion to build and another $150 million a year to run. That produces about 150 gigalitres of drinking water per year. By comparison, the adjacent Powlett River is part of a group of 10 short rivers that drain the surrounding area with a flow of 900Gl/year and Melbourne city (pop 4 million) uses about 360 gigalitres/year. So they'd need three of those $5 billion plants to supply Melbourne with drinking water.
(edit based on Simon's comment below) The cost of desalination is mostly energy - desalination is very energy-intensive. Usually that's electricity to run pumps for reverse osmosis systems, but distillation could be used (has been in the past). The worldwide implications from energy use if lots of people used desalination on a large scale would be significant - those giant solar electricity schemes proposed to supply Europe could instead provide electricity to desalinate... not enough water for North Africa. Energy and water tend to be quite closely linked (most methods of obtaining one require the other), and both are becoming increasingly scarce. If you add a renewable energy plant to feed the desalination plant you're probably adding a billion dollars to the cost. Money that most water-short nations don't have.
That doesn't cover irrigating farmland, which is done on a relatively small scale in Victoria, but in the Murray-Darling basic (Australia's foodbowl) they use about 3500Gl/year to irrigate about 1.4 million hectares. That's 25 desalination plants, going on $3 billion a year in electricity to run them, and the build cost would be astronomical. Obviously using desalination plants to water crops, especially the broad-acre irrigation that Australia uses, is insane. With water that expensive it would be economical to completely change farming techniques to reduce water use, as is done in the middle east and other places.
Many countries use aquifers for irrigation and there are two problems. Often that water is heavily mineralised and sometimes the minerals are poisonous (arsenic for example), but mostly they use them at non-sustainable levels. Either they draw more water from an aquifer than is going in (exceed the replenishment rate) or there's no detectable replenishment and the aquifer is fossil water. Once you've mined out that water, it's gone. Many countries have ongoing problems with wells needing to be drilled ever deeper as the water level drops, or seawater flowing into the aquifer (seawater intrusion).
With numbers like that, you can see why going to war is a attractive strategy (to an economist, anyway). And for people who don't have the money to build desalination plants, that's often the only option. It's likely that one aspect of the current conflict(s) in Syria and Iraq is a contest for control of water sources. There a conflict shaping up between Ethiopia and Egypt over dams on the Nile (Ethiopia will shortly be able to turn the Nile off, which is obviously a concern for Egypt).