What are the best places or types of companies to look for web development internships?
Are there job fairs specifically for web developers? And if not, then what kinds of companies/organizations should I pay attention to at a general regional/city job fair?
In small to medium companies, you're more likely to get a broader range of responsibilities. The more diverse your responsibilities are, the more stimulation your mind and skills get.
Then lastly, there's freelancing or starting your own business. I'm quite biased towards this last option; noting like being you own boss
Big companies are sharp at work and give less learning opportunities. They are very specialized in a limited matter. If you are luckier to find one with your interests, perfect!
Growing companies give you most of the exposure to many things within your criteria.
Even business secrets, and make you dynamic. But they pay less, or you stay shorter.
Web graphics has a high demand. You should determine what part to take in the web development. It is a wide topic ranging from programming, designing, media, online marketing to owning big web based businesses.
I don't know exactly what's the best place to look for web jobs but in general, it's everywhere. At the same time, it can be no where. After I graduated, I was lucky enough to be in web. It was by complete luck. Some did web, some did client application, and some non-programming. Unless you have a lot of experience in web, it maybe challenging to find such jobs. Maybe craigslist may help too. Better chance at smaller companies.
One of the best places to seek out internship work (unpaid or paid) would actually be the businesses themselves. Many web design businesses directly advertise their internship on their portfolio website and therefore if you meet the requirements you can score yourself a placement with some pretty good studios. I also recommend doing the rounds of the freelance job websites (for web design and development), if your portfolio is strong enough you might well be able to skip the internship and directly pick up an entry level position with a good organisation (Places like Krop, Smashing Jobs, Authentic Jobs, etc).
I will reply to UK perspective,
a) Student Placement - like shellstep etc
b) Knowledge Transfer Partnerships - KTP
c) Local E-commerce companies
d) software houses
d) Academic sector jobs for professionals
best of luck