Modern bottom brackets use a sealed bearing type construction that keeps water and debris locked out, so they seldom go wrong. Maintenance is impossible so they need replacing when they do wear out. Older style cup bottom brackets need to be re-greased every now and again, and occasionally will need to have their bearings replaced. There are a lot more teeth on the chainring cogs than those on the rear cassette, so need changing less regularly. Cranks usually a solid one piece construction so there is not much to go wrong.
We like, and recommend, the Crankbrothers Speedier Lever. If you have to pick only one, go with the Pedro’s, but if you want a backup or have some hard tires to unseat, Crankbrothers is a great choice. It’s only one lever (as opposed to a set of two, like the Pedro’s) and it’s longer, so we didn’t pick it over the Pedro’s for portability reasons. But it’s an excellent tool. It has a wide handle you can grip with your whole hand. Like the Pedro’s, the tip is the right size and shape to prevent slipping and stay in place, and the shape of the handle happens to protect your knuckles if you do slip. Even in situations where I didn’t need it, I liked having it because it made me feel like a pro.
Brakes come in a variety of designs depending on the age and style of your bike. Rim brakes offer more modulation but have less stopping power, so are more suited to riding on the roads. Disc brakes can provide huge amounts of stopping power so are best suited to off road riding, although ‘V’ rim brakes are also well suited to mountain biking. All brakes use a friction system to stop the movement of the wheel, so regular checks of the contact pads should form part of your regular cycle maintenance.