Recent research has focused on developing agents powered by large language models (LLMs) to accomplish complex high-level user intents. However, employing LLMs with billions of parameters (e.g., GPT-4) may incur substantial costs on top of handcrafting extensive prompts. To address this, we introduce a Grounded Language Agent for Intelligent Web Interactions, named GLAINTEL. GLAINTEL employs Flan-T5 as its backbone and is flexible in training in various settings: unsupervised learning, supervised learning, and unsupervised domain adaptation. Specifically, we tackle both the challenge of learning without human demonstrations and the opportunity to leverage human demonstrations effectively when those are available. Additionally, we explore unsupervised domain adaptation for cases where demonstrations are limited to a specific domain. Experimental evaluations across diverse setups demonstrate the effectiveness of GLAINTEL in unsupervised settings, outperforming in-context learning-based approaches that employ larger models with up to 540 billion parameters. Surprisingly, behavioral cloning-based methods that straightforwardly use human demonstrations do not outperform unsupervised variants of GLAINTEL. Additionally, we show that combining human demonstrations with reinforcement learning-based training yields results comparable to methods utilizing GPT-4. The code is available at: https://github.com/MultifacetedNLP/Web-Agents-Unsupervised