Espresso machine: A second hand Gaggia Classic can be found on eBay for 100-150 quid, people often throw in extras such as milk frothing jugs and tampers. IMHO this is pretty much the best bang for your buck, considering there is a massive user base and therefore care, repair and upgrade tips and tricks and spare parts are easy to come by. Rancilio Silvia is similar.
Grinder: I went for a Eureka Mignon, and can confirm it is tiny but great! I couldn't find any 2nd hand, so paid 300 quid new. Mazzer grinders are highly rated, but they're bigger and more expensive.
You're likely to find the grinder to be a big improvement, so there is always the option of buying a good grinder now and keep pouring & pressing until you've saved up some more.