MYRTLE BEACH, S.C.—Tropical Storm Hanna picked up speed Friday as it cruised toward the Carolinas, promising to deliver gusty winds and heavy rain during a dash up the Eastern Seaboard that could wash out the weekend for millions of people.

Not far behind was a much bigger worry: a ferocious-looking Hurricane Ike, on a path similar to the one taken by Andrew, the Category 5 monster that devastated South Florida in 1992. Ike could hit Florida by the middle of next week.

In only a few spots in the Carolinas did emergency officials urge evacuations or open shelters for Hanna. Forecasters said there was only a small chance it could become a hurricane, and most people simply planned to stay off the roads until the storm passed.

Rain started to fall early Friday on the Carolina coast, with streets in some spots flooding by late afternoon and winds expected to pick up later in the night. Hanna was expected to blow ashore between Myrtle Beach and Charleston overnight, then race up the Atlantic Coast, reaching New England by Sunday morning. Tropical storm watches or warnings ran from Georgia to Massachusetts, and included all of Chesapeake Bay, the Washington, D.C., area and Long Island.

Some refused to cancel long-set plans to visit the shore. Terry Hash arrived in Myrtle Beach on Thursday, ready to celebrate her 50th birthday with college friends from Colgate University at the Raiders football game against Coastal Carolina.

Several inches of rain were expected in the Carolinas, as well as central Virginia, Maryland and southeastern Pennsylvania. Some spots could see up to 10 inches, and forecasters warned of the potential for flash flooding in the northern mid-Atlantic states and southern New England.

South Carolina Gov. Mark Sanford said people in low-lying areas, mobile homes, camping trailers or places susceptible to wind damage should consider leaving: “Now is the time to look at taking shelter.”