Multi-media communications facilitate global interaction among people. However, despite researchers exploring cross-lingual translation techniques such as machine translation and audio speech translation to overcome language barriers, there is still a shortage of cross-lingual studies on visual speech. This lack of research is mainly due to the absence of datasets containing visual speech and translated text pairs. In this paper, we present \textbf{AVMuST-TED}, the first dataset for \textbf{A}udio-\textbf{V}isual \textbf{Mu}ltilingual \textbf{S}peech \textbf{T}ranslation, derived from \textbf{TED} talks. Nonetheless, visual speech is not as distinguishable as audio speech, making it difficult to develop a mapping from source speech phonemes to the target language text. To address this issue, we propose MixSpeech, a cross-modality self-learning framework that utilizes audio speech to regularize the training of visual speech tasks. To further minimize the cross-modality gap and its impact on knowledge transfer, we suggest adopting mixed speech, which is created by interpolating audio and visual streams, along with a curriculum learning strategy to adjust the mixing ratio as needed. MixSpeech enhances speech translation in noisy environments, improving BLEU scores for four languages on AVMuST-TED by +1.4 to +4.2. Moreover, it achieves state-of-the-art performance in lip reading on CMLR (11.1%), LRS2 (25.5%), and LRS3 (28.0%).