How do I cope with feelings of loneliness and hopelessness?

Loneliness and hopelessness are two pretty strong emotions that are often associated with depression.  Depression can be a short term thing that is triggered by an event (such as loss of a loved one or a break up as in your case), or depression can be a longer-term chronic condition that lasts because of an initial triggering event but then difficulty coping or a series of events that maintain the depression.  Loneliness is not related to the number of people that we know or are surrounded by but is more related to the people (or lack of people) that we connect with and relate to.  Hopelessness is a feeling of intensity that tells us that we don't see an end in sight or feel like things will ever change.  The key to hopelessness is remembering that when we are in a situation or it is recent, we become so focused on being in the situation that we can't see any way out or anything else as possible.  Imagine if we were in a dark room, we probably wouldn't see anything else except the dark. Even when we don't see light at this moment, we need to be able to believe that there is a door or window somewhere that leads to an outside world that has light.

A few keys to remember about your situation here is that just as you pointed out; 1) you recently started grad school which represents a lot of adjustments to possibly a new location, new people, and new routines such as classes; 2) you are probably missing the support of family and friends that aren't as readily accessible, and familiar places that you are comfortable being in 3)  you just had a breakup from someone that you were very connected to, and the most common reaction to that is to feel rejected and question ourselves and our abilities.

The key to overcoming loneliness and hopelessness in your case is multi-layered.  Step 1 is integrating old routines with new ones by looking at what routines and self-care you can maintain despite being in your new environment; this is often basic self care such as eating the same, keeping the same exercise routines, and engaging in the same hobbies or activities such as playing music or sports.  Step 2 is integrating opportunities for new connections while creating opportunities to maintain old connections.  In addition to meeting new people, getting a routine for talking or visiting regularly with old friends and family can be important for support as well as connectedness as it may take time to develop new relationships and build trust.  Lastly, step 3 involves being kind, realistic, and patient with yourself by realizing that you are currently going through a lot of changes and the things you are feeling are normal emotions that demand attention.  All emotions, even negative feelings of loneliness and hopelessness are actually very helpful as they are merely messages that tell us that we need to change something (so we need to pay attention to them and not ignore them and hope they go away).   

Like many things, grad school can be both exciting and scary at the same time because of the many changes, but paying attention to our feelings (just as you are) helps us to adapt to any situation to get the most out of it and hopefully even truly enjoy it.  Good luck with your studies!