I live in Philly and ride "down the shore" several times a year. Was just down there in October.
The most practical option to ride across the river from the city is the Ben Franklin Bridge, but that means riding through Camden. The only other bridge you can cross by bike is the Tacony-Palmyra. That is north of the center of town and leaves you in a congested part of NJ.
What I do is take the PATCO train from center city to bypass Camden and the surrounding towns. Bike are accepted 24/7/365 with no extra charge. Service is frequent at times when you would likely be riding a bike. Fare is something like $3 for the longest trip.
You can also take New Jersey Transit's Atlantic City Line from the city into NJ. In fact, you can take it all the way to Atlantic City. Again, bikes are legal 24/7/365 at no extra charge.
Once you decide how you would like to approach crossing the river, I can give you some route ideas to get you to the shore. I usually take PATCO to the Woodcrest station and start riding from there. Some of the route follows the long-used MS 150 route to Ocean City, diverging at Hammonton, known for its blueberry farms. While the early part of the route is through populated areas (NJ is the most densely populated state in the country), it's a "reverse commute" on weekdays, so it's not that bad traffic-wise.
Even with the train, Cape May is a bit tough to reach in one day. There is a large, decent, large state forest campground in Woodbine that is a good launching pad for the ride to Cape May the next morning. (The way I have been going is 65 miles from Woodcrest to the campground.) From there, how you go depends on what you want to see. Shore towns like Sea Isle City and the Wildwoods with their ocean boardwalks? Straight shot to Cape May via an inland route combined with a bike path for part of the journey?
Personally, if I were not from the area, I would probably go into Sea Isle City, head down what is known as "Ocean Drive" through Avalon (home of the founder of Utz snack foods), Stone Harbor and N. Wildwood/Wildwood. The boardwalk in Wildwood is super kitchy. From there, I might consider heading back inland a bit to Rio Grande to pick up the bike trail to the Cape May area. It was recently lengthened so Google Maps does not show it all.
Again, let me know what you are thinking.
EDIT: Thinking about it some more, I would skip Sea Isle City. Not a boardwalk but rather a concrete promenade, and riding south through town isn't the best. I'd go into Avalon instead, which is the next town down. It's also an easier town to ride into from the west, and there is a bike lane south through Stone Harbor. Either can be reached from Belleplain State Forest with only a few turns.
You also did not mention what time of year you are talking. Could make a big difference traffic-wise. August weekends in Philly can seem relatively empty because so many people go "down the shore."