Inferring meta information about tables, such as column headers or relationships between columns, is an active research topic in data management as we find many tables are missing some of this information. In this paper, we study the problem of annotating table columns (i.e., predicting column types and the relationships between columns) using only information from the table itself. We develop a multi-task learning framework (called Doduo) based on pre-trained language models, which takes the entire table as input and predicts column types/relations using a single model. Experimental results show that Doduo establishes new state-of-the-art performance on two benchmarks for the column type prediction and column relation prediction tasks with up to 4.0% and 11.9% improvements, respectively. We report that Doduo can already outperform the previous state-of-the-art performance with a minimal number of tokens, only 8 tokens per column. We release a toolbox (https://github.com/megagonlabs/doduo) and confirm the effectiveness of Doduo on a real-world data science problem through a case study.