Being rather new to this, I'm aware there are different types of yeast, but not really aware why and for what. The yeast which comes in the kits is fine I guess, but are there any other yeasts to use which could change how the brew tastes?
Ooh, people literally write books on that subject! Many commercial brewers would argue that the yeast is the single most important contributor to the final taste of the beer. There's four main ways the yeast does that :
Attenutation - how good the yeast is at eating complex carbohydrates. Strains that are good at doing that (eg Nottingham) can convert more of the original carbohydrate in the grain to alcohol, but the result is a thinner, drier beer than one made with a low-attenuation yeast like Windsor/S-33 which will have a fuller mouthfeel.
Yeast-generated flavours - mostly esters like the banana you get in some hefes but also responsible for a lot of the fruitiness of English bitter. Perhaps the thing most missing from homebrew versions, particularly dry versions, of British yeasts compared to the production versions you can get from harvesting casks/bottles or from Brewlab. Imperial A09 Pub has most of the marmalade character of the Fuller's yeast, Windsor/S-33 are about the most characterful of the dry yeasts (oh, and the new Verdant one). OTOH, some yeasts (like WLP001/US-05 and lager yeasts) make a big deal of how "clean" they are, they provide little yeast character with the idea of letting the hops and malt shine more.
In general yeast produce more of these flavour compounds at higher temperatures, and not all of them are considered desirable so in fact typical brewing temperatures are some way below the ideal temperature for yeast growth (typically around 30C) in an effort to reduce off-flavours. Lager yeast can ferment cleaner because they are happier at lower temperatures as they have hybridised with a different species of yeast that can tolerate much colder temperatures.
Phenolics - all grain contains phenolic compounds such as ferulic acid, but eg wheat has a lot more than barley. Some yeast like hefe yeast and most Belgian strains have DNA for enzymes that convert the phenolics from the grain into compounds that have distinctive flavours like the clove you get in many hefeweizen, or black pepper. These yeast are called POF+ (as opposed to strains without these enzymes which are POF-) and are one example of how yeast can change the flavour compounds in the grain. Although homebrewers tend to think that only Belgian and hefe yeasts are POF+, a lot of real-world Yorkshire yeasts (but almost none of the British yeasts from homebrew retailers) are weakly POF+, you get a whiff of it in Harvey's for instance (who got their yeast from John Smiths) and Sam Smiths. Also almost all wine yeasts are POF+, which is one reason why people don't normally use wine yeasts in beer - also wine yeasts only eat simple sugars so have terrible attenuation in beer wort.
Biotransformation - yeast can also manipulate chemical compounds from hops, both releasing them from inactive forms and changing them into different and more complex flavours. Some yeast are better than it than others, and since it's not a 100% efficient process some people prefer yeast (like WLP001/US-05) that leave the hops alone as they prefer the greater hop intensity, other people prefer the greater complexity. It's an area that's is the subject of intensive research at the moment, particularly how yeast can release otherwise inactive compounds from hops. But for instance, T-58 can convert Chinook from grapefruit to lime - you'd think it was a different hop altogether.
There are 100+ yeast strains available to homebrewers, so it gets complicated (see eg
this post of mine) but if you wanted to do a small demonstration of the difference yeast can make, make up some wort with 30IBU of bittering hops and say 4g/l of Chinook split across the end of boil, whirlpool and dry hop, and split them into portions with different yeasts :
BRY-97 - classic "clean", fairly high attenuation US yeast (Nottingham is also quite similar)
Windsor or S-33 (or the "mystery silver packet" in most kits is usually the ordinary Muntons one which is a close relative) - somewhat estery, low attenuation
Munich Classic - typical POF+ hefe yeast so should give you some banana and clove even without a high-wheat grist.
T-58 - mildly phenolic so should give a hint of black pepper and should transform the Chinook into lime
Also, I'd like to try a yeast Suitable for coeliacs and those intolerant to gluten, such as what the Chiltern Brewery use, so that a friend of mine can enjoy a beer with me as due to the yeast, Chiltern ales are the only one's he can drink.
As others have said, that's nothing to do with the yeast. Gluten is the main protein in barley but most of it is destroyed anyway by the "normal" brewing process, protein is a bad thing in normal brewing as it's the main component of chill haze. In fact a number of breweries have found that their normal brewing process is enough to get lagers and golden ales below the 20ppm level of gluten that is the official definition of "gluten-free", although they generally don't bother getting it certified as such because of the hassle, cost and paperwork of doing so.
If he's one of the 10% of the population who are gluten-free because of a mild reaction or for other health reasons, he should be aware that there's a lot of beers like that which are not certified but which are under the 20ppm limit. At a homebrew level, if you make pale beer that remains crystal clear at ~0°C then it's probably <20ppm gluten and should be OK for that kind of "weak" gluten-free requirement.
On the other hand if he's one of the 1% of the population that is properly coeliac then you don't want to take any risks, they can get badly ill at gluten levels at even 2-3 ppm or less. Seriously, don't mess about with that.
There's two main approaches to gluten-free. One is to brew with grains such as sorghum that don't have gluten in them to start with - it makes sure there's no gluten but they always seem to taste not quite right, there's a definite taste effect. The alternative is to use an enzyme like the above-mentioned NBS Clarity to destroy the gluten, which is the approach of most commercial gluten-free beers. But it's worth seeing how much chill haze you have in your normal pale beers first, just to assess the problem.