A novel unit-Weibull (UW) distribution as an alternative to the beta and Kumaraswamy distributions with decreasing, increasing, unimodal, and anti-unimodal density shapes has been introduced. When an experimenter wants to eliminate certain test unit groups before the first failure is noticed in all groups, progressive first-failure censoring has been extensively utilized in practice. Assuming that the failure times of experimental units follow the UW distribution, this paper deals with different Bayesian and non-Bayesian estimation issues of model parameters and different reliability indices of the UW model in the presence of data gathered from the proposed censoring technique. Besides the common likelihood function (LF) method, the product of spacing (PS) method is also utilized to carry out the frequentist point and interval estimators. In the Bayesian paradigm, by leveraging the Markov chain Monte Carlo technique as well as the squared error loss, we examined the PS function as an alternative to the usual LF, and both are addressed under the Bayesian setup for all unknown parameters. Using Fisher information gathered from the proposed LF and PS approaches, two different asymptotic interval estimators are created. Additionally, using the acquired Markov chains simulated from LF-based and PS-based, two different Bayes credibility are also acquired. The problem of specifying the optimal progressive censoring is also addressed via four optimum metrics. Simulation experiments are conducted to assess the feasibility of the estimation approaches proposed, taking into account varying group sizes and progressive plans. To show the suggested model's relevance and viability in a real-world setting, two sets of data from the chemical and physical sectors, one founded on vinyl chloride and the other on turbochargers, are examined.