This is a process that should begin early in your undergraduate career. You will need to allow time to become well acquainted with faculty and to provide them with the means to assess your work and your potential for graduate study.
Your first step is to get to know faculty well enough to allow them to write a specific, detailed letter of recommendation that speaks to your potential for graduate study and discusses your work and area(s) of interest. If you don't become well enough acquainted, you may not give the recommender an opportunity to say much beyond 'Angela Smith was a student in my Shakespeare class who demonstrated good writing skills, participated actively, and earned a high grade.' This is not enough information to support a strong graduate school application.
Take advantage of faculty office hours. All UW faculty are required to hold weekly office hours, and they are happy to use this time to speak with you about your plans for graduate study, to answer your questions about the process, to recommend programs, and to discuss their own research. Office hours are a good time for a more leisurely and in-depth conversation -- much better than attempting to chat with professors directly before or after class. See the English Department's people pages to find out where faculty offices are located, and when each faculty member holds office hours.
Consider taking more than one course from a professor whose area of specialization is a good match with your own graduate study aspirations. The ten week quarter doesn't allow much time to get acquainted, and it's difficult for faculty to assess your potential based on one or two assignments.
Strongly consider the English Honors Program. Honors students establish particularly close ties with faculty owing to the small size of the honors cohort and the culminating thesis project, which is completed under individual faculty supervision.
Consider engaging in undergraduate research under the mentorship of a faculty member.
Request letters of recommendation from faculty members in English or in a closely related discipline (comparative literature, American ethnic studies, French literature, etc.), especially if this related discipline is also involved in the lines of inquiry you plan to develop in graduate study. A letter from a professor in an unrelated discipline will be very unlikely to speak to your strengths as a scholar in English studies or your ability to do graduate-level work in English, and will thus not be of much help to your application.
Letters from TAs are admissible if they are the ones who can speak most specifically to your strengths and potential, but it's best, whenever possible, to get at least one letter, and preferably two letters, from tenure-track faculty.
Letters from employers, colleagues, or friends are of no value except in those very rare cases when what you plan to study is extremely well aligned with the work you've been doing (for example, if your proposal for graduate study concerns composition and rhetoric, and you've been working in a writing center, a letter from the center director could be of use). Even so, faculty letters will always carry much more weight than letters from non faculty. Employers and colleagues may know you very well and be able to praise your strengths and commitment, but they are almost never in a position to make a knowledgable assessment of, or a persuasive case for, your potential as a scholar in an English graduate program.
Keep all of the work you have done in English classes, especially from courses taught by the faculty whom you're planning to ask for recommendations. Because there is sometimes a lapse of time between when you work with a faculty member and when you ask for a recommendation, it's very helpful to both of you if you can show the professor the particular papers you wrote in his or her class. This will help to "jog" the professor's memory so that he or she can be specific about your work and your potential, and it will help you to obtain a stronger letter of recommendation -- one that is more detailed and speaks to your particular strengths and interests.
Keeping your work is also important because English graduate programs require critical writing samples. This is typically a paper you wrote in an undergraduate English course that is related to the area of graduate study you are intending to pursue. Many undergraduate classes, at least at the UW, do not require critical papers of sufficient page length to fulfill critical writing sample criteria (which require, on average, at least 20 pages). For this reason, you will want to look over your best college work and begin to expand and strengthen one of your papers, possibly consulting with the faculty member for whom you wrote the paper in the first place. It is not usual for an applicant to "cobble together" two or three shorter papers; this will weaken your application. Graduate admissions committees need to see evidence that you can sustain a complex argument at length.
There is no need to feel awkward about asking for a recommendation. Faculty write thousands of these letters over the course of their careers and are very accustomed to having these conversations. Be prepared to discuss your specific plans and the schools to which you're applying when you ask for letters.
When making the request, give faculty as much notice as possible. A month or two is ideal. It's also a good idea to send them a courtesy reminder about two weeks before your application deadline. (People are busy and deadlines can slip past them. Most faculty appreciate the reminder and do not view it as "nagging.")
When you make your request, it is helpful --and will likely result in a better letter -- if you can provide faculty with copies of work you completed in their courses (preferably including their comments), a copy of your c.v., and a draft of your statement of purpose. This way, faculty will be able to speak in greater detail to your potential and preparedness for specific programs and lines of study.
If you are applying to multiple graduate programs, most faculty prefer that you use a credentialing service so that professors need to write only one letter. The UW Career Center has such a service, called Letters of Evaluation Online (or LEO), to which faculty can submit their letters confidentially and from which you can request that packets of letters be sent to the schools to which you're applying. Interfolio is a similar (non UW) service. These services keep your letters in a confidential file for up to seven years. Graduate programs nearly always request recommendations that you have not seen, and to which you've signed a waiver of your right to review. Confidential letters from faculty are perceived to be more candid. Although you can certainly refuse to waive your rights, institutions tend to take recommendations less seriously when you have done so.
If you are not using a credentialing service, sign the waiver forms included in each program's application materials and provide your faculty recommenders with these forms. It's nice to provide them with addressed, stamped envelopes for each program as well.
Delays are common, and a delay of a year or so is often the best course of action: your application will then reflect your grades in your entire senior year, and you will be more likely to have stronger letters from faculty as well as a more well developed critical writing sample. You will also have more time to reflect, research, and develop a stronger application packet if you are not trying to rush through it during the fall quarter of your senior year (most graduate programs have deadlines in December and January for a program that begins the following autumn).
If you are planning on a delay, it is a very good idea to ask for your letters of recommendation before you graduate, while the details of your work are still fresh in your professors' minds, asking faculty to direct them to a credentialing service (see above) to be kept on file until you're ready to apply. If the gap between your undergraduate and graduate study has lasted just a few years, it's possible to ask faculty to write an updated letter, but it's not absolutely necessary.
If you have taken a longer break between completing your undergraduate work and applying to graduate programs, securing letters can become more difficult. If you already have letters on file with a credentialing service, do attempt to reconnect with faculty and ask for updated letters. If you do not have letters on file, obtaining them after a long break can be especially challenging. You can first attempt to reconnect with former professors, but this is not always possible owing to retirements, relocations, and so forth. It's also possible that the professor may remember you only vaguely, if at all. If a former professor does agree to write for you, you will likely need to spend some time discussing your work and your plans.
You should also bring your knowledge of the English studies discipline up to date by reading scholarly journals in your proposed area of study; applicants who have taken a break of five years or more often sound "dated" in their statements of purpose, especially those applying in fast-moving and fast-changing areas, such as gender theory or cultural studies.
If it's not possible to secure letters from former professors, you'll want to plan to take some additional upper-division English courses before you apply to graduate school so that you can update your knowledge and establish new relationships with faculty. In most cases, you can take these courses in non matriculated status. Again, letters from employers or colleagues are of little to no value to graduate admissions committees.