As part of my PhD research, I am using the SWAT model for hydrological simulations at the watershed scale. Per se, the only calibration method available with SWAT is the manual calibration, which can be accomplished within the ArcSWAT (ArcGIS) interface (or rather QSWAT for QGIS). However, the process can get tedious and you are prone with lots of errors if you are starting with SWAT.
I would rather suggest using the SWAT-CUP alternative. Basically, SWAT-CUP is an automatic calibration program for SWAT model. The workflow is quite simple: you start preparing your model set up with ArcSWAT (or QSWAT), and then you feed this set up to SWAT-CUP. There you can run sensitivity tests on selected parameters, or perform model calibration and validation using many optimization algorithms: SUFI2 (Sequential Uncertainty Fitting), PSO (Particle Swarm Optimisation), GLUE (Generalized Likelihood Uncertainty Estimation), ParaSol (Parameter Solution), MCMC (Markov Chain Monte Carlo). You are not required to know the in-depths or the maths behind such algorithms as the program let you use them in a breeze and produce the necessary graphical outputs to evaluate the simulation runs. The latest version of SWAT-CUp is available for free (https://www.2w2e.com/home/SwatCup). If you buy the license, you can even use parallel processing features.
I can also suggest a swath of papers that might be relevant to evaluate your model calibration.
Article SWAT: Model use, calibration, and validation
Article Model Evaluation Guidelines for Systematic Quantification of...
Article Hydrologic and Water Quality Models: Performance Measures an...
Article Hydrologic and Water Quality Models: Performance Measures an...
Article Characterising Performance of Environmental Models
Article A continental-scale hydrology and water quality model for Eu...
Article A Guideline for Successful Calibration and Uncertainty Analy...