In low wind areas, a prop with higher solidity may work slightly better than a fast low solidity one but this is marginal and in normal winds a 3 blade prop with tsr from 4 to 7 usually gives the best results and has a much lower thrust on the tower. As this type of prop suits direct drive up to at least 17ft diameter it is the most commonly used. All transmissions have losses so it is hard to beat direct drive in the region where it is practical.
In the early days all commercial size turbines were forced to use some form of gearing, but even those now have reverted to direct drive.
If you are in a marginal wind area, where things are not strictly viable ( average wind speed below 7mph) you may find a benefit from a higher solidity prop with more than 3 blades and a direct drive alternator becomes big and costly and a small ratio speed increase with a chain drive could help. I would only consider a chain, not belts or gears, and to get any efficiency and life out of it the chain needs to be enclosed.
Mary B mentioned the chain drive alternators used by Chris Olsen. Chris was using ferrite magnets and for the same speed a ferrite alternator will be about 4 times the size of a neo one. By doubling the speed this lets him use a much smaller and efficient alternator. It does in this case make sense but remember that Chris is a first class mechanical engineer with all the design knowledge and facilities to build a reliable and efficient chain drive. To work it needs first class engineering and will not be cheap.
Most of the machines here are designed so that they are easy for home build with limited facilities, other options are certainly possible. The radial flux alternator is almost certainly better than most axial machines but doesn't lend itself to home build as it needs concentric parts and has no tolerance for design errors. If you get the turns wrong with an axial you can change the air gap, with a radial you rewind it.
When you mention large speed increases, yes they can be done, the old water pumper type mill with multi blade fan can produce electricity with a high ratio speed increase, but contra to your expectation the result will be very poor compared with a similar diameter 3 blade direct drive and will have large wind loading problems in high winds.
Many things are possible and in experienced hands may work fine, but the designs devised by Hugh Piggott and the Dans of this board have been perfected so as to be easy to build, give good performance and good reliability, I would stick to something similar as a first step into wind turbine building.
Flux
