What do I need to get started with American style Barbecue?

My recommendations would be as follows.

First, buy a cooker that will do everything you need, but is priced reasonably. You will want something that fits the following criteria:

1. Provides direct grilling capabilities.
2. Convects heat fairly evenly, so that you can use it for low-temperature, long-time, indirect-heat cooking (what most people in the US would consider barbecue).
3. Has the capability to control temperatures easily.
4. Has plenty of cooking area.

The basic Weber 22.5" (57cm) kettle grill fits all these needs, on a modest budget at that. You can set it up for high heat grilling, low heat barbecue, and just about anything in between, just by how you arrange your fuel. When you close the lid and arrange all your fuel to one side of the chamber, the parts of the chamber that are not over the fire heat very evenly, allowing you to cook things like pulled pork and brisket. The vents make it easy to control temperature by limiting or expanding the flow of oxygen to your fuel. Finally, there are numerous after-market accessories available to if you decide you want to expand your horizons, such as cast iron grates, pizza stones, inserts that make your grill into a pizza oven, etc. So you can start off with a small investment, and then build on piecemeal with accessories. I would recommend the 57cm model, as the 47cm one makes it difficult to cook with indirect heat. Link here: http://www.weberbbq.co.uk/product/one-touch-original/

Once you have your tools, make these sites your bibles:

• http://tvwbb.com/
• http://amazingribs.com/

Both are full of great information for making good food on the grill.

This is exactly how I got started. Eventually, after I felt proficient enough with my Weber grill, I purchased their Smokey Mountain smoker, as well as a couple of additional grills. It's become an obsession, and one that I have been able to get into fairly cheaply.

While certainly more expensive, the ProQ smoker that you link to has the advantage of versatility. As @SAJ14SAJ points out 'grilling' and 'barbecue' are often used interchangeably. If you remove the middle section of the ProQ you can use it for grilling, if you leave it in (as pictured) you can smoke with it. Again, leave it out and put a suitably sized pizza stone on the grill and you have an excellent pizza oven.

Something like the classic Weber kettle grill (per your link) is certainly well suited for grilling and can be used for smoking. To accomplish smoking using the Weber you need to properly bank your fire to one side, manage your cooking temperature (to keep it below @120°C (250°F)) which means keeping hot coals external to the Weber and adding them at the appropriate times. You should also add one aluminum dish to hold water to introduce steam in order to help regulate heat and prevent the drying of your meat. None of these consideration would I recommend for the 'beginner'. I can't speak to your budget, but I believe a beginner will get more out of the ProQ (or similar) equipment.

The "Chimney" starter is a nice accessory as it replaces the need to you use lighting fluid (either bottled or presoaked into your charcoal). Either way, you should let your coals reach a hot white ash stage before you begin grilling. This will achieve a consistent heating surface (rather like pre-heating your oven) and ensure that any and all starter fluid has burned off (if it was used).

The Food Network's US website (I believe they have a separate UK site) has a collection of videos from Bobby Flay, Alton Brown and others called "Grilling Essentials" which you should find informative and entertaining. Beyond that I suggest you search out other Bobby Flay videos on grilling.

There are really three separate things that are encompassed by "American Style BBQ" plus your description.

1. Grilling - using medium to high direct heat to cook your food. Temperatures range from 350F (175C) to 600F (315C). For steaks and pizzas, you can also get in to very high temperatures 700F + (370C), but you're going to find that impossible at an entry level.
2. Smoking (short) - using low temperatures and smoke to cook your food. Temperatures between 200F (93C) - 350F (175C). Time less than 4 hours.
3. Smoking (long) - Same temp and technique as short, but times range from 4 - 12+ hours.

For all of these, you will need:

1. Grill tools - A spatula, fork, tongs, grill brush. You can use regular kitchen ones, but make sure their long so you can work safely with the hot grill and that they're metal (don't use a rubber spatula!). The grill brush is important to clean the grill before using.
2. Heat source - charcoal for any of them and wood for smoking. Get hard lump charcoal preferably. Do not, under any circumstances, use lighter fluid in this process (in the coals or adding it).
3. Starter - you need to light the coals to get started. A chimney is a great, cheap way to do this, but is not required. If you're using hard lump charcoal, you can dip paper towel in vegetable oil, nestle it in to your coals, and light it. It's cheap and easy, but might not work with normal charcoal. There are also electric heat elements that work great but are a bit gadget-y.
4. Cooking Implement - If you want to smoke primarily, then a dedicated smoker can have some advantages to a grill (like access to coals, better setup to pull smoke across the food, etc). The question is, which will you do more often, grill or smoke? You can definitely do both on Webber grill. If you want to do steaks and pizzas, you need to check that you can a) get enough coals to get proper temp and b) get your food close enough to the fire. If you go with a grill and want to smoke, get a grill grate that is hinged so that you can add more coals / wood to the fire without moving your food.
5. Thermometer - if you are smoking, you need a thermometer in your grill / smoker so that you can properly check grill temperature. You don't need this for grilling.

The distinction between long and short smoking is important for the gill / smoker. The longer the smoke time, the more work it is with entry level equipment. Still possible, but much more effort.

If you ever get serious about both grilling and smoking, I highly recommend A Big Green Egg (or any of it's competitors).

There is a backyard barbecue, and then there is pit barbecue. Webber BBQ grills are for backyard barbecue - you are cooking meats and vegetables over open flames atop a metal grid. High heat, fairly quick cook times - sausages are great for this, especially if they're par-boiled first. There is an art and science to this, to put a decent char on the food without burning it or leaving the center raw. You need to learn when and how to turn the items, how to prepare them for the flame, and how to move them to an area where they'll receive heat indirectly to finish cooking.

For this, you'll need:

• Kettle grill
• Starter chimney
• Decent quality charcoal
• Crumpled Newspaper (to start the charcoal in the chimney)
• Barbecue tools - traditionally a large flipper, tongs and meat fork.
• Quick read thermometer

Pit barbecue is cooking meat through indirect heat and smoke, very slowly. This is how you'd prepare pulled pork and babyback ribs. This can be a very difficult cuisine to master - you need to understand how to prepare the meat for smoking and how to pair it with the correct sauce.

To get started, you'll need:

• Smoker (Alton Brown and the internet shows you how to build your own on the cheap)
• Quality hardwood chips/sawdust to produce the smoke
• Large tongs or meat forks to turn the meat.
• Quick read thermometer

BBQ is actually a hotly debated topic in the states, no pun intended. In several parts, BBQ refers to any outdoor grilling event, in which case you might want to know about all the different things that can be cooked over coals. These should be called cook-outs. There's little science behind grilling beef patties and sausages, it's easy enough for college graduates to understand. I'm sure Brits enjoy this as much as Americans do. However, in the heart of BBQ culture, BBQ is nothing like cook-outs. I'll assume you refer to the refined, Southern technique of slowcooking smoked meats.

First the hardware: If you have a lot of money, the best device for the job is a Traeger grill. However, there are ways of fashioning smokers out of basically any container (I'll echo the recommendation for Alton Brown's "Grilling Essentials" linked by @CosCallis). The key is being able to maintain a constant, low temperature for a very long period of time.

Then the software: A good BBQ sauce serves as a marinade, a basting solution, and a finishing sauce. These always have robust smokey, salty, sweet, sour, savory, and spicy notes, adjusted to the preference of the griller. Adobo, mesquite, worcestershire sauce, ketchup, apricot jelly, apple cider vinegar, soy sauce, tabasco sauce, honey, beer, and bourbon are all really common ingredients that are concocted for a homemade BBQ sauce.

Another important aspect of your software is wood for smoking. Applewood is probably the most popular smoking chip for BBQ, followed by mesquite and cedar. The wood adds a tremendous amount of flavor and aroma during the smoking process.

Temperature is really important. The philosophy of "low and slow" is essential to BBQ. Beef and pork can be smoked at 140*F (60*C) for as long as 18 hours to melt connective tissue and give a thick pink smoke ring in the meat as deep as 1" (a couple cm) the deepness of this pink ring is a crowning achievement of good BBQ. Poultry needs to be smoked at higher temps to kill salmonella. Fish can be cold smoked.