Make Le Régent Montmartre Hostel at the foot of Sacré Coeur your headquarters. It's an easy shot on the Metro to major tourist attractions and the gay nightlife in the Marais or around Les Halles, but there's plenty to keep you occupied in the neighborhood.
After a free breakfast at the hostel, walk to the nearby Anvers Metro and hop a train eastward towards Nation. You're headed to Père Lachaise Cemetery, but get off at the Gambetta station instead of Père Lachaise for an easier walk. Spend the morning ambling among the tombs of the late great celebrities like Oscar Wilde, Jim Morrison, Colette, Gertrude Stein, Edith Piaf. The funerary sculpture throughout is museum-worthy.
Returning to the world of the living, backtrack by Metro or foot to Belleville, an artsy working class neighborhood whose Tuesday and Friday street market (8 a.m. -1:30 p.m.) is more like a souk than a French marketplace. If the market is open, pick up street food or the makings of a picnic. If not, walk uphill on Rue de Belleville, exploring Asian bazaars along the way, to eat inexpensive homemade dumplings at Restaurant Gui Xing.
Check out the open-air (and free) display of graffiti murals on tiny Rue Dénoyez. Along the semi-pedestrian street are art galleries and bars, including the popular Aux Folies, whose terrace is busy day and night.
Take the Metro back to Anvers. That ticket will also get you a free transfer to the funicular, the little train that ascends to the highest point in Paris - the gleaming Sacré Coeur Basilica. You can see all of Paris from the basilica's terrace. Inside under the impressive dome are equally impressive mosaics.
Peckish? Pick up a take-away crepe at Au Piche du Tetre and nosh as you browse the artwork displayed on the Place du Tertre. Watch the sunset from the Sacré Coeur terrace before walking down through the park, past the antique carousel, to your hostel.
Relax in the game room with a little foosball before considering the evening ahead.
Stop in La Fourmi for a cheap draft (or two or three) and affordable daily menu and see if anything is scheduled at Divan du Monde, a music club across the street. Then check out the after-dark panorama from Sacré Coeur and - weather willing - join a United Nations of young tourists and musicians on the steps.