Historian in training at what level? I would suggest taking a historiography class, as that class will likely teach you more than you can learn here about how to evaluate sources.

Historiography is concerned with the mechanics of studying history; it provides a rubric, a lexicon, and reference points for the methodology, sources, historical theories, and other relevant tools for exploring a topic. These influences are meant to be visible, and are meant to clarify the specific methodological vector a historian used while researching and writing, specifically to announce its position within the wider historical conversation.

Historical theories are descriptions of a certain methodological approaches to study, and are meant to compliment or challenge other theories, especially when they are used on the same topic. For instance, a study of 19th century medicine might come to different conclusions if one author is studying it from a broad Marxist perspective - focusing on the macro-economics of medical services and how they are influences by ideas of class structure of the time - and another is studying it from a military perspective - looking at the efficacy of treatment and its influence on the armies and campaigns in question. Both of these hypothetical works can provide valuable insight onto the topic, but also come to opposed or divergent conclusions. They might not even be concerned with the same questions!

The way to navigate this is simply to be familiar with the methodological framework used by both authors. The introduction or abstract of both works should lay this out explicitly; more often than not, it will situate itself in the historical conversation as the author understands it, and it will lay out not only the historical question and the suggested answer (its thesis), but also the limitations, chronological and methodological, that qualify its conclusions.

There is quite simply no replacement for understanding the broad sketch of the historical conversation, and being fluent in historical theory and methodology. Bias is not something to eliminate, it's something to qualify, because no work of history is ever meant to be the single perfect work on any given topic; it's just meant to be a meaningful moment in the continuing conversation.

This certainly sounds jaded, but if you're going into graduate studies (and often even as an undergraduate) it's difficult/impossible to read every page of every book you'll be assigned. There are ways to get a lot out of the books while still saving time! Grad students I know often call the process I'm going to describe "gutting" the book:

You start by doing some background research on the author. Is this their first book? Does it stem from their dissertation? Where did they study and who was their dissertation adviser? If it's a second book or later, how does the project relate to their other works?

Next, read the introduction carefully. A good book will tell you all you need to know in the introduction. What's the main argument? What sources does the historian use to make their point? Is there a certain methodology they use? And perhaps most importantly, what previous works are they dialoguing with? I'm not totally clear on what you mean by "bias"; however, if you can identify the historians argument and how this either refutes a previous argument or adds to the existing knowledge, you'll be better able to understand exactly why the book was written and published. Also pay attention to the chapter layout as the historian will normally explain the main arguments of each chapter in the introduction.

I would next read the conclusion. Seems out of order, but in the conclusion the historian often walks through their smaller arguments and shows you how they relate to the larger argument. From there you can decide how much of the book you still need to read to understand the big picture. Book reviews also help a ton especially if the book is outside your field of speciality.