Salem, MO for the upper Current River; Van Buren, MO for the lower Current River (and Doniphan, MO for the far lower Current River). Eminence, MO for the Jacks Fork (and central Current River); and Alton, MO for the Eleven Point River. You can use these cities in your GPS to get you to the general area and to use as your jumping off point to explore the region.
The nearest commercial airports are Saint Louis, Memphis and Springfield, MO.
The Missouri Scenic Rivers area is best reached via Interstate 44 from the north or route 60 from the south. From I-44 take State Route 68 south from St James, then take State Route 19 south into the scenic rivers region. You can also take State Route 72 south from Rolla to Route 32 east, then hit State Route 19 in Salem and take it south or take State route 19 from Cuba through Salem south. All three State Routes will get you to where you want to go. You will come upon the Current River first when heading south on hwy 19 ( there are multiple access points to the river sections that a map would better serve you to get to where you are wanting to access the river at if the Current River is your destination ), and continuing south you will happen upon Eminence, MO which is the main city along the Jacks Fork. Traveling even further south you will eventually happen upon Alton, MO which is the main city for the Eleven Point River area.
From US Route 60, you will reach State Route 19 in Winona if you come from the west. You can then take SR 19 north to access the Current River or Jacks Fork (or go south on 19 to get to Alton, MO and the Eleven Point River), or continue east to State Route 103 to visit Big Spring near Van Buren. The Ozark National Scenic Riverways Park Headquarters and Visitor Center are in Van Buren. If you follow US 60 from the east, you'll reach SR 103 first and then SR 19 about 20 miles further on. If you are coming from the southwest, then 60 will also lead you to Van Buren, MO and the south portion of the Current River. Journey on to Winona to access hwy 19 north to the Jacks Fork or south to the Eleven Point River area. Taking 160 east out of Alton will take you to Riverton and one of the main access/take-out spots on the Eleven Point River.
Main roads throughout the scenic rivers region are well maintained state highways. These are hilly and winding, so be careful and take your time! Watch for bicycles, horses and logging trucks. Roads into backcountry areas are typically dirt and less maintained. Some are suitable for passenger vehicles and RVs, some are not. Please inquire locally before heading off on dirt roads in the area. A dirt road that was fine last year may be muddy or even washed out due to a recent storm. Ask at ranger stations, sheriff's offices, canoe rentals or local businesses for information.
I have put together a concise listing of things to see and do in the Missouri Scenic Rivers area (link below). The most popular floating stretches for the Current River are the Cedar Grove to Round Spring stretches on the upper Current River. For the Jacks Fork, Alley Spring to Eminence is the most popular stretch. For the Eleven Point River, Greer Spring to Riverton is the most popular stretch.