As has been said, a lot depends on your budget.
It would be feasible to include five Open courses in this trip - Hoylake, Birkdale, Royal Lytham, Turnberry, and Troon - but most would not consider that great value for money!
I would pick from West Lancs, Formby, St Annes Old Links, Seascale, Windermere, Silloth, and a couple along the Ayrshire coast which is so far an area I have not visited. Formby and Silloth are must-sees.
Don't miss the opportunity to explore the wonderful Lake District for at least an afternoon and early evening - perhaps after a morning at the quirky Windermere before a full day at Silloth the following day.
Find lodgings in the Keswick area rather than in Silloth itself!