Wireless federated learning enables many Internet of Things (IoT) devices to perform collaborative learning under the coordination of the base station (BS). Besides, the BS is also in charge of providing data transmission services for a lot of communication-centric tasks such as environmental monitoring and sensing. However, the limited bandwidth available at the network edge makes it challenging to support massive connectivity and frequent model exchange between the BS and IoT devices. Therefore, it is of significant importance to improve the spectrum efficiency of an edge network having both learning- and communication-oriented applications. To this end, we propose a simultaneous federated learning and information transmission (SFLIT) framework to overcome the communication bottleneck while improving the learning performance in resource-constrained wireless networks. Specifically, the proposed SFLIT framework allows the concurrent transmission of model parameters and raw data over the shared multiple access channel. First, we derive the optimality gap of federated learning to quantify the impact of wireless communication on the convergence performance. Then, a long-term optimization problem under time-varying channels is formulated to minimize the aggregation error of model gradients while satisfying the quality-of-service requirement of information transmission by jointly optimizing the receive and transmit beamforming. Finally, simulation results demonstrate the effectiveness of the proposed SFLIT framework, and show that our algorithm is able to reduce the aggregation error efficiently under different parameter settings.