I'm a recent grad from university in game design ... something in the fields of software design/engineering, modelling and simulation or general programming.
Unfortunately the generic "game design" schools tend to produce graduates for roles that are sparse. Read the forum FAQ for a lot of details. We routinely (and regrettably) see too many people who get a useless "game design" certificate, where a company is just in the business of selling dreams and separating people from their money. Hopefully you went to a reputable school that taught you real skills.
For the role, pick ONE. Studios don't say "We have an opening for a modeler/designer/programmer, must have 3+ years experience in each field."
Game designer. This is not entry level. You might get a lower job, like a level designer. A portfolio for a level designer would include views of levels and the reasons behind what you did and why you did it. Explain why things are where they are, and not just "I plopped down some barrels and crates". Did you give every location a similar risk/reward situation? Describe things like how you avoided choke points and overly-defendable positions. Provide flowcharts for collection games or progression-based games, detailing the progression and why the design is good. This should be old-hat if your school actually studied real game design. The portfolio should include a wide variety of levels, preferably written with actual tools like Unity or modding systems.
Modelling. This can be entry level. Show models you have done. Good modelers can make anything. Era-specific buildings and vehicles, clothing, barns, and fences. Know rigged characters; people, mechs, animals from zero to eight legs, sometimes with wings. Modelers also spend a lot of time on environmental objects; show that you can make rocks, trees, grass, hills, mountains, and whatever else is needed. If your only models are elvish women and guns, your portfolio is incomplete.
Software engineering, simulation, and general programming. Prove that you can do them well. Usually this is done by actually writing game code in your own example programs, but can also be more of an academic focus like showing off your A* algorithm in one place, showing off a simulator or a text adventure, or otherwise demonstrating that you can actually make games. The more evidence, the better for you and the shorter your job hunt will be.
Be local. If there are no game studios near you, be prepared to move first. It is exceptionally rare for for a studio to pay to relocate an entry-level developer.
Don't expect to break in with no effort. There are a very small number of game studios relative to other industries. It can take months or even years to break in, so be prepared to apply for jobs in similar fields and start your career. If you cannot immediately get a job making games but you can get a job elsewhere in entertainment, or in programming medical imaging software, or in something else that can help you build your portfolio, do so.
