Let $F_n$, $W_n$, and $\widehat{K}_n$ be the graphs obtained by joining a vertex to $n$ independent edges, a cycle and a path of order $n-1$, respectively. In this paper, we give new bounds for the Ramsey numbers $R(F_n,F_m)$ and $R(W_n,W_n)$, which improve those due to Chen, Yu, and Zhao [EJC, 2021] and Mao, Wang, Magnant, and Schiermeyer [G&C, 2022], respectively, and establish lower and upper bounds for $R(\widehat{K}_n,\widehat{K}_n)$. Moreover, we present a blow-up technique to establish some new lower bounds for the Ramsey numbers of wheels versus cliques.