The keyword "web design" is a very competitive term. Achieve ranking for such keyword will depend on many factors: domain age, domain page rank, size of website, your current position, etc. Hence, it might take you months or years to rank for.
In general SEO is about on-site and off-site optimization. Both are important, you should start with optimizing on-site first and this include: internal linking structure, keyword rich content.
Create a web page and write a unique article (700 words) focusing on the keyword "web design". Include keyword in Title, first sentence, last sentence. Keep keyword density at 1%, this means using keyword once every 100 word. Include keyword in H1, H2, H3 headings. Bold your keyword once, italic once and underline once. Add an image to the post named with your keyword and include keyword in image alt tag.
Try to link internally from other pages within your website to this page using keyword as anchor text. After you're done with that, start off-site optimization.
Now you have to start building high quality backlinks from other websites using keyword as anchor text to your optimized web page. Find authority websites with high PR and get links from them. Try to diversify your linking method, don't use just one method.
Sources of links:
Article directories.
Press release.
Blog commenting.
Profiles.
videos.
Social networks.
Social bookmarking.
You should spend 20% of your time optimizing on-site and 80% for off-site optimization. Again, such keyword is very competitive and you should do proper keyword research before you even start building your site. I know it's an internship and the website is already established, but you can find other profitable keywords with low competition to work on. Instead of wasting your client time, suggest them to target an easy keyword which they can rank for in a couple of months instead of years.
SEO is not magic, it's a long term strategy. You can't rank this keyword during your internship period.
Hope this helps.