If you are going to use an ice bath to chill the wort, you don't want to do full boils. It'll take forever to chill 5 or 5.25 gallons of boiling hot wort that way. You'd want to do a partial boil of 3 or so gallons (most kit instructions are written assuming this method), chill the reduced wort in an ice bath, then when chilled down to 80 degrees, mix it with chilled water in your fermenter to get it down to pitching temps (preferably below 70).
The basic decision is one of equipment investment and complexity. Consider how much you want to spend on equipment. Also consider space to brew and store equipment.
Mucho dinero: To do full boils, you'll need more equipment, and you'll be brewing outdoors. Buy a propane turkey fryer with a 40 quart pot minimum, stainless if you can find it but aluminum is fine, and an immersion chiller with garden hose fittings. You may be able to make an immersion chiller cheaper than you can buy one if you like to DIY. Immersion chillers are simple and effective, low maintenance, easy to clean. A plate chiller is another option, usually faster chilling, but you'll definitely want to drill your pot and put in a ball valve and pick up tube if you go with that method. Plate chillers require a more rigorous cleaning and sanitizing routine than immersion chillers; having a pump to circulate PBW through it after use is handy, and in my opinion almost a necessity.
Inexpensive: To do partial boils then topoff in the fermenter, buy a 20 quart pot. You can brew on your stove top. An immersion chiller with kitchen sink fittings is very welcome, but you can use an ice bath in the sink to chill the 3 gallons of wort you end up with in the pot at the end of the boil.
A lot of people here will tell you to buy everything up front to do outdoor full boils. I usually don't. It's fine if you have the space for it and are willing to spend that much money to get started. You're probably going to eventually outgrow it and replace it, so why spend all that money up front? The advantage of starting simply and cheaply, which is what I always recommend, is that you can start brewing easily with little expense and complexity. For many people, that facilitates more frequent brewing. There are many ways to expand to full boils, partial mashes, or all-grain brewing. You can be cranking out and drinking kit after kit doing partial boils on your stovetop while you study the options and plan for your dream system.
Here is the link to the online version of John Palmer's
How To Brew. Note that the online book is the first edition. The hardcopy book you have is probably the third edition.