I've read that grad schools really like it if you have tutoring experience (since grad students often teach) or some kind of departmental service (which could be volunteer tutoring or otherwise mentoring lower-division students, or volunteering at conferences, etc.). I've found that any activity where you can hang out and chat with grad students is also useful because they can give you advice about grad school; in fact, if you start talking to a grad student, within five minutes they'll probably start giving you advice about grad school whether you want it or not. For example, when I volunteered at a department conference (setup, cleanup, food service, etc.), one of the grad students told me that getting your own external funding (she has NSF money) helps you get into grad school, because departments will be happy that you're already partially funded, since that means if they admit you they can give you less out of their budget than they would if you had none of your own funding at all. I'm not really familiar with the application process for that, but if you have enough time for that, I'd recommend looking into it.
External funding aside, attend department events (colloquia, conferences, etc.) even if you're not volunteering, and even if you won't get explicit advice. It'll give you a better understanding of what academia is like, what sort of activities will be expected of you, what the unspoken cultural norms are. Lots of students apply to grad school with a very uninformed, unrealistic idea of what it's like. If you can experience some of the real daily life of academia, you'll be less likely to come across as naive or unprepared (not that you necessarily would anyway, but there are just so many who are, and you need to convince admissions committees that you have a good idea of what you're getting yourself into).
The LSA Institute is (according to my professor who's taught there, anyway) potentially very useful if you have grad school plans, because it has undergrads and grad students and professors from all over, so you can network and get insider advice about different schools. You might conceivably be able to get a recommendation out of it, though I'm not sure how big the classes tend to be (and of course it is brief). But you can take classes in areas that your school might not offer, which will give you a better idea of what exactly you want to pursue in grad school.
But of course you can also read up on different areas on your own; obviously the Institute is really expensive and there are plenty of other, more affordable activities. Basically grad schools are primarily interested in your ability to do research, so, like the other poster said, prioritize getting research experience, whether it's assisting a professor or doing a thesis/independent study project (or both). If you have a choice between research experience (even really menial stuff) and interning at some sort of vaguely linguistic organization, the latter is better for your employability but the former will likely be better for your grad school admittability. Sometimes professors will take you as an assistant if you ask, even if they don't advertise a position, but don't be discouraged if you don't get anything on the first try. (I applied for a research assistant position with a professor who knew me and I never even got an interview, and then a year later the same professor emailed me out of the blue and offered me a better position than the one I'd previously applied for, so you never know.) Also: can you code? If so, do they know you can code? If not, can you learn? Everyone can use a coder, so professors will be more likely to want you, and if you can't find a professor who'll take you, well, you'll still have acquired a widely applicable and marketable skill.
If you can't get anything with a professor, you might still be able to get something with a grad student. Of course working with a professor is better because it's generally inadvisable to have a recommendation from a grad student alone (though if a prof can co-sign it then it should be fine), but working with a grad student will still be valuable experience and perhaps a way to get your foot in the door. One time I was sort of aimlessly chatting with a grad student and she offered to let me work on stuff with her (and it sounded interesting, too, though I ended up not taking her up on it); grad students are an often overlooked but potentially effective way to get research experience.
If you can't get anything in your department, try related departments. Often there are linguists in specific language departments, or psycholinguists in psych departments. If you have an education department, they can use linguistics students too sometimes (especially if you pitch it right, since so many people don't really know what linguistics involves). Or obviously, for you, computer science.
Of course, if you're in computational linguistics, you might not necessarily be planning to stay in academia forever (and even if you are, there's a lot of industry comp ling jobs and very few academic jobs at all, so, you know, that might happen), in which case employers will probably want to see some non-academic experience. Either way, it doesn't hurt to take a year or a few years off from school after undergrad and get some work experience; in fact it could make you both a more competitive applicant and better prepared to handle graduate school successfully.
If you have specific schools in mind, and if there are any professors in your department who went there for grad school or used to work there, try especially to get to know them. I know several students from Undergrad X who got into Grad School Y after having worked closely with undergraduate advisors who got their PhDs from Grad School Y, and I don't think it's a coincidence.