Sequential recommender systems (SRSs) aim to suggest next item for a user based on her historical interaction sequences. Recently, many research efforts have been devoted to attenuate the influence of noisy items in sequences by either assigning them with lower attention weights or discarding them directly. The major limitation of these methods is that the former would still prone to overfit noisy items while the latter may overlook informative items. To the end, in this paper, we propose a novel model named Multi-level Sequence Denoising with Cross-signal Contrastive Learning (MSDCCL) for sequential recommendation. To be specific, we first introduce a target-aware user interest extractor to simultaneously capture users' long and short term interest with the guidance of target items. Then, we develop a multi-level sequence denoising module to alleviate the impact of noisy items by employing both soft and hard signal denoising strategies. Additionally, we extend existing curriculum learning by simulating the learning pattern of human beings. It is worth noting that our proposed model can be seamlessly integrated with a majority of existing recommendation models and significantly boost their effectiveness. Experimental studies on five public datasets are conducted and the results demonstrate that the proposed MSDCCL is superior to the state-of-the-art baselines. The source code is publicly available at https://github.com/lalunex/MSDCCL/tree/main.