There's a reason logo design is one of the most expensive types of design. It seems even more ridiculous when you consider how small and simple a logo can be, but ultimately, it's about the process. Designing a logo is neither simple nor quick. If you want a good result, you can't rush and you can't skimp on any step; whether it's research, scamping, or even refining a vector curve. Each designer will naturally have his or her favoured method but most will start with the basics: researching the company or brand and sketching out ideas on a piece of paper. Both of these steps are crucial because they create the foundations for the final design. For example, scamping and sketching is the ideal way to get as many ideas down as quickly as possible before you forget them or lose your inspiration. Most of these ideas will be discarded but somewhere in the mess of doodles, a design will stand out.
Of course, generating these ideas is another story entirely and that's where research comes in. How can you design an effective logo if you know nothing about the company? Or worse, you design the perfect logo but it's too similar to another logo. Research is primarily finding out information about your client's company but it's also finding out what other logos are out there, especially those related to the same industry. You need your logo to be relevant to the industry that the company belongs to but you also don't want it to look like all the others. You'll almost certainly have noticed this about logos from certain sectors; you can spot financial logo from a mile away for example.
Once you have your doodle, you then need to develop it further. This is usually when you'll hop onto the computer and start drawing it in a vector-based program such as Illustrator. Unfortunately, this can sometimes push you a step back or make you rethink your approach because what looks good on paper, doesn't always work when it's recreated digitally. I would suggest only moving to the computer when you have 3 - 4 scamps which you think are promising. Not only will this lessen your chances of having to go back to the sketching stage but you'll also have more than one option to who your client.
I tend to first draw up my logos in black so I can concentrate on the shape and feel rather than the colour palette. That said, most clients won't want to see the first drafts in just black and will ask for a colour or two even if it's not the final choice. I'd advise caution here; never show your client an option that you don't want them to choose. Nothing ruins a good logo quicker than an ill-matched colour palette. If you're not good at colour matching, find a palette generator online or simply search for an existing colour scheme that you like the look of.
Equally important to the colour is, of course, the font you use (provided you need to use one). Again, a good logo can easily be ruined by choosing the wrong font. Sometimes, your client will insist on a certain font and often there is very little you can do about that, but if you can, only show them options which suit the logo style, and of course, the company it represents. There are thousands of logos, free or otherwise, available online and many sites have tutorials on how to apply these fonts effectively.
As a designer, you'll already know that there is a certain amount of instinct involved in all designs and this applies to logo design as well. They won't all be winner unfortunately but designing a logo is an art and as with all art, the more you do it, the better you become.