What are current ways in which astronauts and spacecrafts are protected from radiation? Are their any new ways/ideas to limit the amount of exposure to radiation?
So currently we don't actually do anything to protect them from radiation (if we exclude the basics like UV shields and being able to alter the attitude of the craft to place the highest density of metal/solar arrays/whatever in the direction of the incoming radiation) As Paul mentioned, limiting the amount of time spent in space is pretty much it. NASA have researched the use of various shields such as water etc, but the mass cost is still prohibitive.
Those astronauts who have spent long terms on the ISS are extensively studied to understand the impact on them, but they aren't protected.
The ISS orbits between 205-270 miles (330-435 km) above the Earth at an inclination of 51.64 degrees. That's low enough that it can get by with minimal (read: basically no) shielding, because Earth provides all that it needs. For deep space settlements, roughly 7 tons of water per m^2, or 11 tons of lunar regolith per m^2, is required to shield something outside of Earth's protection against every type of radiation. The Van Allen belts themselves stretch from ~400 miles (650 km) to 36,000 miles (58,000 km), so a habitat located below the belts can use minimal shielding, while one above would require the full protection (and thus considerable added mass, potentially in the millions of tons). The International Commission on Radiation Protection sets the limit for yearly exposure to radiation at 20 milliSieverts, and for pregnant women, the limit is 6.6 milliGrays (Grays are a measure of radiation absorbed). Polyethylene is also an option for radiation shielding, and is slightly better than water.