I've been told by numerous people that (1) my undergraduate university will be disinclined to bring me on as a graduate student, and that (2) it's a bad idea to attend grad school where you completed your undergraduate degree, anyway. Is that true? If so, why?
- If you stay at the same school (this applies even more when you join grad school immediately out of your undergrads), it'll be a matter of remaining in your comfort zone - same department, a faculty who know you, even the same apartment/neighborhood! This can be a major factor, depending on the person concerned - the pros of staying at your Alma mater are all about convenience IMO.
- If your UG department has an influential professor with whom you've worked before and are planning to continue as well, that can be very advantageous - as having such a faculty get to know a student's work as an undergraduate can lead to a very strong recommendation (since he has accepted you in the grad program, it is reasonable to assume your work had impressed him during your undergrads).
An important advantage of going to another school is that you will be exposed to a completely different department, with faculty who may have diverse research ideas for you to work on. The department, in turn, will benefit as well as a new student from another school will cross-pollinate their department with fresh ideas. This is so important that some top universities have a strong bias against accepting their own undergraduate students into their graduate programs.
As Henry points out in the response to Artem's answer, many graduate departments—especially at top schools in the US—have a "no undergrads admitted from our department" policy. Both my undergraduate and graduate schools have adopted such policies.
In general, unless you have a very strong reason for staying at your undergraduate school—either a personal situation, such as a spouse who has a job in the area, or the opportunity to work on the world's only "X" (whatever "X" is)—then you are much better served by going somewhere else for graduate school. You will have the advantage of working with new people, plus you avoid the very strong stigma attached to having all of your educational pedigree at a single location.
I will address the two points separately:
If you have a strong application, then your undergrad university will be more inclined to take you as a graduate student. The reason is that to have a strong application, you need very good reference letters. The reference letters are probably from professors at your undergrad school, and thus they will carry a lot of weight there (compared to other schools where your former supervisors are less known). Thus, it is often easiest to get into the school you graduated from.
I've heard the second point myself, and I actually advice it/try to follow that advice, too. The reasoning behind it as that at the school you graduated from, you already know everybody. Thus, if you stay you will continue to work with the same people and won't meet new collaborators. If you go to a new school, you will have to meet new people and expand your network.
Staying at the same place is very uncommon in the US, where there is a presumption against it. It does not mean that it is impossible or never happens, but it counts against you in admissions.
This is mostly true when referring to doing undergraduate and PhD studies at the same place. Normally it's fine to do your Masters and undergraduate at the same place, or to move between undergraduate and Masters and then stay at the same place for PhD.
The situation is very different in Europe. In fact, professors may attempt to recruit their best undergraduates to work for them as graduate students and this is seen as a positive thing for all involved.
There is an increasing stigma against hiring faculty that has no experience outside of a given university (derisively called in-breeding), but I do not think that it applies to undergraduate->graduate transition.
Others have answered (2) in more detail.
Here's an intermediate step that some of my friends took: they stayed on to do their Masters in the same lab where they did their undergraduate study, then moved college (and country) to do their PhDs. That way, they got more research experience in an excellent lab in their home country (keeping costs down a bit), then used their PhD stipends to offset the cost of living abroad.
The only advantage I see in entering graduate school in the same university where you completed your undergrad program is having a continuity between your undergrad research programs and graduate research.
Typically it takes some time for a new grad student to get accustomed with research activities, with the routine of a new lab or research group, getting to know your advisor, etc. Also there are some adjustment of your personal life: you are probably moving to another town, choosing another home, finding a new favourite grocery store, a new gym, etc, etc. This takes time and certainly impacts how productive you are.
If you are enrolled in undergrad research programs, and you're already well adjusted to your lab's/research group routine and workflow, it might be that you'll feel much less of an impact of changing from undergrad to grad research programs. It might be that you can start being productive right away because you don't have to worry about a lot of things.
On the short term, it might be that you manage to turn this into one extra article published at the end of your PhD. It's not improbable. But it's not incredibly probable either.
You incur in a lot of risks if you don't change. If you spend too much of your formative years working on the same research group, you risk becoming too well adjusted to its workflow and research programs, to the point that you can't see alternatives.
Also, the real impact of a few months of advantage and an extra article, if it really happens, is probably offset by the advantages of moving to another university. A couple years after your PhD the time you lost finding an apartment and banging your head against the wall to understand your new research program will most likely be forgotten and will not influence your career at all.