Researching where and what firm you would like to work with is an essential first step in landing a design internship. There is no point spending time in a company in a city which does not appeal to you or in a company that does not really relate to what you want to do. Look in books, magazines, and websites to scope out which designers and companies work you like. If you are at university or have just finished it may also be worthwhile asking your professor where best they think that you would fit in and what company would be suitable. An objective view is often more reliable.
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From this research make a list of maybe 20 companies you would like to contact, rank them so that you know where to really put the best effort in, but also be realistic. You may want to increase the list if at first you are unsuccessful as securing an internship can be a very hard task as there is lots of competition.
Before you contact a company make sure you know all the essentials of the company, their newest ventures, their most successful work, the designers and their projects, maybe even where the designers studied and worked previously. One of the most important details to get hold of however is the person who to send your work to.
Make sure you introduce yourself with a brief yet concise letter of introduction. Include details of where you study and what subjects or what you have previously studied. If you have completed other relevant work experience, also include this. State why you are contacting them and mention what appeals to you about their company in particular. If the company is far from where you currently live, make mention of the fact that you are willing to relocate and understand that the position is unpaid. Your contact details are very important, state which is the best way to contact you but don´t be shy, some people prefer email but someone may want to pick up the phone and have a quick chat with you so do include your telephone number as well.
Check your online and offline portfolios
Make sure that you have an accessible online portfolio, either your own website, LinkedIn, Twitter or professional Facebook account that any prospective employees can see and find out more. Also consider joining an online portal such as Behance or Carbonmade as these sites are designed specifically to showcase emerging talent, many of these sites also have a job board so it will be easier to search for opportunities.
Offline, make sure you include as much as your work as possible, even if you have only just a small selection of work, this will enable companies and designers to see what you have done but also to evaluate your potential. Also include a CV with a bit more detail about work experience and your experience with specific computer programmes. If there is a particular area that you excel in make sure you put emphasis on this. Then grab the details of the person in charge of hiring, send this package of along with your introduction letter and wait for the good news!
If you have heard nothing for a week or two contact the companies by email or telephone and ask to speak to the person who you sent the package to. Confirm receipt of the package and ask if they have had time to see your work, if so ask if they have any questions and be prepared to have the answers. By being persistant and following up on your first application the company can see how dedicated you are. If you still hear nothing from a firm, keep trying again after you have more work in your portfolio. Your persistance will pay off