As an important class of spiking neural networks (SNNs), recurrent spiking neural networks (RSNNs) possess great computational power and have been widely used for processing sequential data like audio and text. However, most RSNNs suffer from two problems. First, due to the lack of architectural guidance, random recurrent connectivity is often adopted, which does not guarantee good performance. Second, training of RSNNs is in general challenging, bottlenecking achievable model accuracy. To address these problems, we propose a new type of RSNN, skip-connected self-recurrent SNNs (ScSr-SNNs). Recurrence in ScSr-SNNs is introduced by adding self-recurrent connections to spiking neurons. The SNNs with self-recurrent connections can realize recurrent behaviors similar to those of more complex RSNNs, while the error gradients can be more straightforwardly calculated due to the mostly feedforward nature of the network. The network dynamics is enriched by skip connections between nonadjacent layers. Moreover, we propose a new backpropagation (BP) method, backpropagated intrinsic plasticity (BIP), to boost the performance of ScSr-SNNs further by training intrinsic model parameters. Unlike standard intrinsic plasticity rules that adjust the neuron's intrinsic parameters according to neuronal activity, the proposed BIP method optimizes intrinsic parameters based on the backpropagated error gradient of a well-defined global loss function in addition to synaptic weight training. Based on challenging speech, neuromorphic speech, and neuromorphic image data sets, the proposed ScSr-SNNs can boost performance by up to 2.85% compared with other types of RSNNs trained by state-of-the-art BP methods.