How you decide to sleep and the gear you decide to carry is completely up to you. Now just because you are not under a hand-woven canvas with ropes made from things you found on your trail doesn’t mean that you are not doing bushcraft. Many people prefer to sleep in a tent. That said one of the most popular methods of bushcraft camping is sleeping under a tarp - This can include using hammock or sleeping on the ground - a tarp is a great option for a shelter that covers a large amount of ground when compared to the space that it takes up in your pack.
Tarps are a great shelter option when it comes to sleeping in a woodland environment. With some knowledge of a few basic knots and an abundance of trees to tie off to you can be very easy to pitch. Tarps are also very inexpensive. You could use a basic tarp from a builders merchant - it might not be cool and camouflage like some of the higher-end tarps on the market but this could provide a very cost-effective option for someone looking to get into bushcraft camping. With a tarp and some guidelines in your pack you have everything you might need to set up a basic shelter in the woods - you don’t even have to carry tent pegs. These might be useful if you want to make some different kinds of shelters but are not a necessity)
The next thing you are going to need is going to be a decent sleeping bag (rated for the kind of environment you are going to be sleeping in). Some people will prefer to use a woollen blanket (a more traditional bushcraft style), however, if you are just starting out and getting used to sleeping out under a tap a sleeping bag is the best option. You can invest in a good bad that will pack down small and will be very lightweight - Down sleeping bags are good for these reasons. If you are going to be sleeping in the warmer months then you might look at something like an ex-army summer weight bag. These can be picked up from a local surplus store for very little money and compress down very small in your pack.
If you have decided not to use a hammock you will need to make sure you insulate yourself from the ground - This will not only keep you warm and prevent heat loss but this will also increase the comfort of your new bed in the wilderness. There are a number of options for this such as a sleeping pad or foam roll matt. You might want to take into consideration the weight and size of the solution you have chosen as you will, of course, need to fit this on or in your pack. You should also think about using a bivvi bag. This is basically a windproof and waterproof bag that your sleeping bag goes inside. This will keep your sleeping bag dry (from both rain and morning/evening dew) and also warm by trapping air inside and around the internal sleeping bag. Another benefit of a Bivvi Bag is that it will also protect your sleeping bag from dirty that will (over time) degrade its performance.
Your end goal might be to sleep out using very little and making improvised shelters. Our advice would be to initially stary by sleeping out for one or two nights with your tarp and then progress to using just a sleeping bag, bivvi and sleeping matt. As you gain more confidence and your skills increase you will then be able to create weatherproof thatchings, self-feeding fires and raised beds - Once you have the skills your imagination is the limit!