The troubleshooting process always starts with identifying what works. If the problem is power related (whether battery or a question of the laptop not turning on) the first step is establishing that power is getting to the laptop. This means checking that the LED on the transformer brick is lit, and if it isn't (or doesn't have an LED), that it's plugged into a good power outlet. You can check that by unplugging the transformer and simply plugging in a lamp. Some of the oldest notebook models have an internal transformer, so the line power (110 VAC in the U.S., 220 most other places) goes directly into the laptop body. The next question is whether or not any of the little LED status lights on the laptop light up with the power plugged in. Even the oldest models usually have a power good status light. If you have positive power status and the notebook simply won't turn on, the next check is the battery. Some models of notebooks will not operate without a good battery installed, but most will, so Google up your particular model with a search like "operating without battery" and find out if your laptop will operate with a dead or missing battery. If the battery isn't an issue and the laptop still won't turn on one with the power good status light lit, it could be a switch failure, but it's more likely a power regulation or mainboard failure. Troubleshooting power regulation or the motherboard requires test equipment or spare board to swap out.

The guts of a laptop are crammed into such a small, cramped space, that the cooling fan is absolutely critical. A replacement laptop fan and heat pipe should cost well under $50, you may even get by with a generic fan replacement for a few bucks, but the job is fairly involved and differs from manufacturer to manufacturer. I don't get excited about noisy laptop fans, I had one in my Toshiba Satellite that got noisy within a year of my buying it and continued noisy for the next four years without failing. On the other hand, you don't want to wait until you get heat damage to replace the fan. If the fan gets increasingly noisy over time or starts noisy (and slow) then quiets down as it picks up speed, I'd replace it at the first opportunity. Assuming you've owned the notebook for a while, you should be familiar with how long the fan usually takes to come on and how long it runs. If the fan never comes on, unless you're working in a freezer, it's probably dead..

Fortunately, laptop hard drives are the one really generic part (aside from most memory) that you don't have to worry too much about replacing. Depending on the model, you may be able to really upgrade to a much bigger drive on a replacement, but you probably won't get the benefit of a faster interface on an older notebook and the BIOS may not recognize most of the capacity, so there's no point in spending much more than you have to. Laptop hard drives can be extremely easy to replace or moderately difficult. The difference lies in how they are accessed. Many older notebooks allow you to replace the hard drive through a single-screw access panel on the bottom of the unit, sometimes it's right under the battery or the RAM. Other laptops require that you crack the body open, remove the keyboard or the motherboard (assembly varies from manufacturer to manufacturer), really take the whole thing apart. The interface for the IDE cable on the drives that come out easy is often fixed in place, so the drive basically plugs in, while the drives that require you to take the whole thing apart often make remove the connector on a flexible (and fragile) flat cable before removing the drive.

Modern laptops are all sold with built-in wireless adapters. Some power notebooks used by corporate road warriors will have powerful cellular adapters that connect the laptop to the Internet via the cell phone infrastructure, but the standard built in adapter is an IEEE 802.11b/g wireless transceiver that allows the laptop to connect to local wireless routers and access point within a hundred feet or so. The signal strength is proportional to both the distance between the router and the notebook, and the stuff inbetween, like doors, people, walls, etc. That said, the main hardware issue you'll run into with a wireless Internet connection is the wireless adapter in the laptop being switched off! The vast majority of the time, Internet connectivity problems will be due to operating system software settings or router security settings. The easiest way to troubleshoot whether your connectivity problem is dependent on your local network is to take the laptop somewhere else and try it.