When going to the Gulf coast straight west on SR 44 you'll arrive in the Crystal River - Homasassa Springs area. This is the southern end of the part of Florida called 'Big Bend', it's the curve from Panama City Beach around and down. Around the bend is mostly marsh and natural perserves and the little fishing town of Apalachicola is about the only town of size.
To reach a sandy beach, there are little spots here and there in Cirtrus County, but for the most part you need to turn south on US 19 and go into Hernando County near the town of Spring Hill and turn west on Hwy 50, or take I-75 south and exit Hwy 50 westbound, but you will have traffic going through Brooksville. Stay on route 50 and you will end up on a state park called Pine Island, not to be confused with the Pine Island near Ft. Myers. It's a beautiful beach and has facilities but not much in the way of commercial businesses.
Pine Island
You can continue south about 15 miles to Hudson and the first real beaches that are attached to the towns. In Hudson turn right on Clark St. and go straight ahead to Hudson Beach. It's a small beach used by the locals, I use to live in Spring Hill and went many times. It's has nice seafood/bar right on the beach playing Jimmy Buffet and Bob Marley on the speakers. However, it's not much, here's photo.
Hudson Beach
Be advised, Hwy 19 is a nightmare of non-stop businesses, malls, restaurants, so don't be in a hurry. To get to real beaches with motels, marinas, restaurants, etc you'll want to go to Clearwater Beach. Continue south through New Port Richey, Holiday, Tarpon Springs, Innisbrook Golf Resort, Palm Harbor, Dunedin, and then Clearwater turn right or west onto Hwy 60 and follow to the beach. This is a beautiful beach with everything, but can be crowded. Another way to go if you're not in the sightseeing mood is take I-75 south to Tampa and join I-275 into Tampa and then near the Tampa Int'l Airport look for Hwy 60 Courtney Campbell Causeway and get on that and take it all the way west.
Clearwater Beach
Once you're on the beach and if it's too crowded and you continue south and find beaches for the next 25 miles, they're will be Indian Rocks Beach, Reddington Beach, Maderia Beach (check out John's Pass Boardwalk while you're there), Treasure Island and then St. Petersburg Beach.
St. Petersburg Beach
Also there is a really neat state park called Ft. Desoto. It's a huge island and is located south of St. Pete Beach, you'll need to take Pinellas Bayway (small toll) as if you're headed back into St. Petersburg but then take the exit to Tierra Verde.
Fort Desoto State Park
Now you can skip Citrus, Hernando, Pasco and Pinellas Counties and all the beaches above and just stay on I-75 and go around Tampa and go to Bradenton and Sarasota and the beaches that others have recommended such as Siesta Key, St. Armands Key, Anna Maria Island and Long Boat Key. However the area around Sarasota can be more expensive just like areas further south around Naples.