Welcome aboard, Tijl!
A backpack, a suitcase, and a guitar would not cause any problems, especially if you are travelling outside of the busiest periods. You will not be the only student, Erasmus or otherwise, to be making the journey from Paddington to Oxford, and many of the others will be similarly encumbered. If you can get all of your kit on and off the train quickly and without assistance, and can stow it in the luggage racks, you are unlikely to raise eyebrows. A piano, grand or upright, would be frowned upon. As John R helpfully mentions, there are general conditions of carriage, often honoured more in breach than observance, but common sense is generally sufficient. If you have so much luggage that it needs three trips to get it all off the train, you may find the doors locked and the train on its way before you have finished, and you may never see your kitchen sink again. There are no charges for carrying personal luggage - this is the railway, not Ryanair.
The type of train can make a big difference. In
the timetable, you will see that many of the services have a letter 'H' in a circle above the times. These signify high speed trains, usually of eight carriages, and with more space for luggage than some of the other offerings. Sadly, what is shown in the timetable is not always what shows up at the station, so you may want to allow yourself the flexibility of a ticket that allows you to get any train, rather than one specific service, although that may come at a greater price.
First Great Western offers a luggage forwarding service, with an option especially for students. Details can be found
here. I have never used this, and have no idea of how good it is, or what it costs.
Enjoy the "City of Dreaming Spires", and don't forget to travel west whilst you are in the
UK▸ . Most foreign students (I assume that, as an Erasmus scholar, you are from overseas) visit only Bath, Stratford on Avon, and London, so missing some of the spectacular jewels in the crown that can be found in this Septic Isle.