In addition to supervised classification, soil and NDVI indices will help you to discriminate barren land from prepared agricultural land. Sometimes you may need to do a change detection for different seasons so that you can differentiate planting period from harvest period.
Landsat images allow you to use both supervised and unsupervised classification (both tools available in Arcmap and QGIS). In the former, you will make an expert delineated signature file, which gives you the possibility to include different types of agricultural land. This is important because different agricultural types will have different vegetation and thus a different reflective signature. With unsupervised classification, the software will group the landsat image into a certain amount of groups. This technique is purely based on differences in reflective signature, and thus makes a very distinct grouping of classes, which is good if you have very distinct differences between the land cover classes. However, it might not be suitable for diverse agricultural systems, especially if you have grassy or tree crops. In those case I have experienced that the unsupervised classification has difficulties distinguishing between natural land cover and agricultural areas (for example natural wetlands vs. rice paddies or advocado plantations vs. forest). It thus really depends on the diversity and characteristics of the land cover in your study area.
It is also important to realise that both techniques will always return a certain amount of erroneous results. While a human specialist can for example detect agricultural land use by the artificial boundaries of the plot, both supervised and unsupervised classification only use the reflectance. The key to good science is to 1) minimise the error by for example combining supervised classification for the major reconstruction, with independent checks with unsupervised classification and Google earth images. You can subsequently manually correct the detected errors. And 2) include land cover validation and model sensitivity analysis.
I added a paper I published recently on a very complex study site with a detailed methodology on how to overcome some of the challenges.
You can use Supervised Classification, where you should select Maximum Likelihood Classifier algorithm for extracting agriculture land. Presently, Landsat OLI/TIRS is best for classification as it provides 30 meters of spatial resolution. You can also do the same with Sentinel data which provides 10 meters but you can't get older data here (before 2010).
The best way is phenology based classification technique. As you want to use Landsat imagery, EVI & GVMI along with NDVI/GNDVI is the best suite indices in this regards.
On the other hand, Sentinel 1 is also very very useful to identify Agricultural land/Crop land. You do not need to worry about data issue. Google Earth Engine is there to help you to derive outputs without downloading a single SAR image.
In this case the following YouTube video is enough to achieve desire outputs.