Sharpening is (in theory) a large and complex topic. However, if you're just getting started in woodworking with handtools, the vast majority of your blades will be reasonably long, rectangular and sharpened straight across. This type of blade includes all bench planes, block planes, rebate and plough planes, and "normal" chisels, which is enough tools to do rather a lot.
In bevel down tools (bench planes) the bevel angle (is believed to...) only effect edge strength - as long as it's lower than the bedding angle, and thus provide "relief".
Many learned discussions have been held on which exact edge angle is required for which task in which material with which blade steel. If you want to know about all that, find a non-basic reference on sharpening.
Back in my basic world:
Since everybody wants to make an informed decision, here's a thumbnail description of the major abrasive choices on the market.
Since they've been around a while, oilstones have the advantage of being available second hand (and thus cheap), and the process of using them leaves nice rust preventing oil on your tools. Using them is (however) messy, and you run the risk of getting dirty oil in places you don't want it, like on yourself, or your woodwork.
A waterstone tends to cut more rapidly (for a given particle size) than the other abrasives because they wear away quickly, continually revealing fresh, sharp, abrasive particles. However the same rapid wear results in dished stones, which need flattening. The lubricant (water) is also prone to invoking rust if you're not careful about drying your tools after sharpening.
For the purposes of this page I recommend the cheap side-clamping jigs, which (here in the UK) are referred to as "Eclipse Style" after their best known (and now sadly defunct) manufacturer. These jigs will allow you to sharpen the blades this page is about; other jigs are better (though more complex and/or expensive) for more unusual blades.
Final honing, using fine abrasives is carried at the desired bevel angle. This is called the secondary or honing bevel. The difference in the angles means that (at least to start with) the fine abrasives are working only at the tip of the primary bevel, on a small surface resulting in relatively rapid progress. As time passes, the secondary bevel gets larger, until the coarse abrasive is once more used to reestablish a (nearly...) full length primary bevel.
Double bevel sharpening can be achieved with any of of the honing techniques listed except for (2). The secondary bevel is deliberately small, and there's no way you could balance a blade on it.
I reckon a Norton combination stone, a natural fine stone, a side clamping jig and some oil can be snarfed up at a decent car boot sale for around a fiver.
Knock up a simple projection gauge, and use single bevel sharpening at the angles I recommended earlier. The first time you do this on an existing blade you'll spend a while on the coarser ("medium") side of the combination stone. This is a once-off activity. To re-sharpen a tool, all you do is:
Now just move the tool backwards and forwards on the stone, working over the whole surface to avoid wearing the stone in any one spot. Continue this until a burr or catch can be felt on the back of the blade. This is easily detected by running either your finger tip or finger nail over the edge. As long as you haven't let the blade get too blunt, 15-20 strokes should be enough.
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