welcome to MCN's forum Vlad
You have the option of direct access and getting an unrestircted bike; but if confidence is an issue this may do more harm than good. Insurance will be, err, memorable too.
125s are great, but generally offer little that a 250 won't (other than learner legality). Don't know what your budget is but a Ninja 250 or CBR 250 (NOT the 80s/90s model!!!!) would be a wise choice. Insurance will be managable you can commute, get decent MPG, add hard or soft luggage, get decent speed up on roads for fun riding, overtake more safely and cope with motorways just fine.
I know all this because i graduated at 18 from my Yamaha SR125 to a Kawasaki GPX250 and did all my travelling by that bike - daily commute, fun, and regualr triangles from my parents in Middlesbrough, to Uni in Staffodshire, to girlfriend's in Essex. GPX did it all in its stride.
If you can't go new 250, look for decent used Ninja 250s or be picky about GPX or ZZR 250s (they'll be a little aged by now). Some dealers may have stock of "grey import" VT250s (like a Duke Monster). If you can find one, a decent ZZR or RF 400 could be an outside choice - not as big a step up as a CBR/GSXR/ZXR/VFR 400.
If you like cruisers - a Rebel 250 or Dragstar/Virago 250 may be just the bike for you...happy hunting