Yup, for Old Street, you're best getting the Thameslink train, not the Gatwick Express. These will be headed to Bedford or Cambridge (and slightly slower ones to Peterborough). Pro tip: the very rear carriage has a first class compartment which is declassified at present (there will be a notice on the cycling display to tell you this).
You can change at London Bridge for the Bank Branch of the Northern Line (yes, it has two branches in central London - long story). It's quite a big station, with a fair old walk to the Tube, but there are lifts to get you to all levels. Old Street has a few steps then an escalator.
An Uber from Farringdon (another stop on the Thameslink line) is also a possibility.
If you really don't want to walk, and it's a weekday, you can get a Peterborough or Cambridge Thameslink to Finsbury Park, and take a Moorgate train from there to the very creepy commuter station at Old St (hidden away from the Tube platforms). There's a fair few steps to get out, though, and not many lifts at Finsbury Park.