Head west, following signs for the Barbican Centre. There is a yellow line on the floor to show you the way. This takes you straight to the main arts centre.
After the arts centre, check out the Conservatory. It is free, and worth at least a quick look inside. You have to go right through the Barbican arts centre & then go up a floor. If you are a gardening fan you will want to stay a while.
After the Conservatory, retrace your steps back to the Barbican Centre entrance and then head south. You want to be on a walkway called 'Gilbert bridge', which takes you past St Giles-without-Cripplegate. Then walk along 'The Postern', and on to the 'Alban Highwalk'.