As online music consumption increasingly shifts towards playlist-based listening, the task of playlist continuation, in which an algorithm suggests songs to extend a playlist in a personalized and musically cohesive manner, has become vital to the success of music streaming. Currently, many existing playlist continuation approaches rely on collaborative filtering methods to perform recommendation. However, such methods will struggle to recommend songs that lack interaction data, an issue known as the cold-start problem. Current approaches to this challenge design complex mechanisms for extracting relational signals from sparse collaborative data and integrating them into content representations. However, these approaches leave content representation learning out of scope and utilize frozen, pre-trained content models that may not be aligned with the distribution or format of a specific musical setting. Furthermore, even the musical state-of-the-art content modules are either (1) incompatible with the cold-start setting or (2) unable to effectively integrate cross-modal and relational signals. In this paper, we introduce LARP, a multi-modal cold-start playlist continuation model, to effectively overcome these limitations. LARP is a three-stage contrastive learning framework that integrates both multi-modal and relational signals into its learned representations. Our framework uses increasing stages of task-specific abstraction: within-track (language-audio) contrastive loss, track-track contrastive loss, and track-playlist contrastive loss. Experimental results on two publicly available datasets demonstrate the efficacy of LARP over uni-modal and multi-modal models for playlist continuation in a cold-start setting. Code and dataset are released at: https://github.com/Rsalganik1123/LARP.