Is a PhD in a small university in a small country restrictive for an academic career abroad? I am located in a small European country and studying computer science in one of its Universities, admittedly not the most prestigious one. I am set to finish my undergraduate degree at most a year from now. Recently, an opportunity has come up as I have been offered a PhD position right after I am done with my bachelor's. Along with that, I have been offered a position in a EU funded project that is right on my field of interest.
I am struggling to make the choice as I would have to give an answer before I am done with my degree so that if I decline they will have the time to look for someone else to take the spot.
My purpose and hope is to have an academic career in a university that does significant research in the field I am passionate about. However, those universities even in Europe seem to be the most prestigious ones (Oxford, Edinburgh, ETH Zurich etc). I can currently boast a few conference publications, some upcoming journal publications, and hopefully two more will come out of my undergraduate thesis currently underway. All in all I hope to have at least 8 papers to my name by the time I get my bachelor's. I should specify all my works but one are research focused, not surveys.
My dilemma is as follows:
Do I choose to do my PhD in my university and hope to break in the academic world abroad with my post doc?
Do I do a master's elsewhere and then hope to get funding for my PhD abroad?
Or can I hope that I will get accepted for a PhD position abroad if I apply after my undergraduate is done? This would be the optimal scenario but I have not come to contact with any cases that did their PhD right after their undergrad, much less in a different institution. I guess I am wondering what my chances of pulling this through would be, and if they are low to zero, which of the aforementioned two scenarios would help me more to pursue an academic career in the Computer Science field?
Thanks for any insight and advice.
This is important career advice and should be given by someone who you trust who knows you personally.  If no one comes to mind, ask your existing professors in the field.
As you suspect, there are powerful reputational and network effects from prestigious programs.  It would be hard to argue that your employment prospects after graduating Cambridge are fairly good! You have heard of it and respect it. Your question is - how does your future employer feel about your Uni?  Have they heard of it? Do they respect it? Do they teach what you want to learn?
The question is very specific to your field.  Perhaps your Uni is universally known for the subject, or has undeniable geographic advantages - like studing Marine Biology at U-Hawaii or being across the street from CERN. It is hard to tell how strong that reach is until you have dipped your toes in the job market.
Note: while you can trust your teachers to guide you honestly, do not expect them to talk poorly of their current institution to a student. They will not.  Simply be aware that if the local institution &quot;is good enough&quot; is not the same as saying &quot;we are one of the top institutions in the world, of course you should stay.&quot;