A side note should be made about items marked with the bare letter 'K'. A single letter is not trademark-able. As a result any kashrut organization can use it. You usually can call the company and find out more about their certification to see if it is alright (K is often used by local rabbis who do not do certification fulltime). It is best to use a regular hechsher otherwise, only if not such product can be found should you use this. It should be noted that many Rabbis have started Vaads, and their standard may not be yours (a good example recently seen on the shelfs, kosher for Pesah candy with corn syrup).
The rule of thumb to follow is first use a hechsher. If you can not find the product with a hechsher, use one with a 'K' (after checking the contents). Finally, if all else fails, you can check the ingredients. It be noted that by conservative standards products with 'K' (which is not verified through the company) and products verified through contents are not considered safely kosher. As a result, using these should be done with care, and you must check with the Rabbi before bring any such product into the synagogue kitchen.
Because of the nature of kashrut, most Vaads are Orthodox. This is economic. Certification is also big business. Vaads usually charge a percentage of sales for each item bearing a hechsher. There is not enough extra business to justify a Conservative hechsher, (as most Conservative Jews will rely on an Orthodox hechsher, the only added business will be Conservative Jews who keep kosher). Another result of the large cost of hechshers is that there are products out there that are inspected but do not carry a hechsher. This is because the company believes it is not worth the costs to carry the seal (many Vaads charge for carring the seal on top of the inspection) but worth being inspected. Hershey chocolate and Sunshine cookies are two good examples. The only way to discover these is through the kosher trade publications. There have also been scandalous behavior by various organizations when important accounts moved from one organization to another.
Fresh fruit, vegetables, milk, sugar, flour, pure juices (except grape in certain circumstances), coffee, tea, and other "pure" products. Canned vegetables canned in water need no hechsher (although one is preferred). Canned vegetables in syrup, tomato and blended juice, sauces, and other processed canned food require a hechsher. So does all prepared food. Also things used with food, such as dish soap, aluminum foil, etc. need hechshers. It should be noted that raw vegetables should be inspected and cleaned of insect life.
Be aware. Non-dairy does not mean pareve. Many non-dairy products use milk derived cassin.
Hechshers without an accompanying mark are usually pareve (except where obvious like ice cream). Hechshers on milchig products usually have a 'D' for dairy. On fleishig products they usually have a 'M'. 'P' usually means kosher for Pesah. Every once in a while you will see a 'F'. This stands for fish. Some Orthodox Jews follow a practice of not eating fish and meat together (from a Talmudic aside that it might not be healthy). You sometimes se ME or DE meaning meat equipment or dairy equipment (but otherwise parve).
Checking kashrut by ingredients is not a perfect way to do it. First, many chemicals have animal origins and have questionable kashrut. Also, nonkosher products can be used in processing and not show up on the label (like lard used to grease pans).
Things to watch out for when checking the contents. Look out for whey and other milk products in bread. These breads can not be used with meat. Shortening must say vegetable shortening, otherwise it has animal shortening and is not kosher (not that vegetable shortening guarantees kashrut, often animal based shortening and emulsifiers are added for stability - and these will not always be listed on the ingredients.). I refer you to page 97 of your book for a more complete list of chemicals.
One special exception is gelatin. This product is of animal origins. It is made of the bone, which is not considered a food in many respects. As a result, the CJLS has ruled that gelatin, in its processing, stops being a food and is kosher. The Orthodox are mixed on their opinion of this.