To be on the safe side I would suggest that you can assume that anything published before 1900 is now out of copyright. Anything 'newer' than that should be treated as potentially still in copyright until you can be sure it isn't. The general rule is that copyright lasts for the lifetime of the author plus 70 years after their death. So, for instance, in the case of a young author who wrote his first book at the age of twenty and then lived until he was 70 years old, copyright in that book would last for 120 years (50 + 70).
From this you can see that where a book you want to use is less than 120 years from its first publication, you will need to try and find out when the author died. If they were reasonably famous this usually isn't too difficult, but with more obscure or foreign authors you may need to be more resourceful. And bear in mind that illustrations within a book will attract a different copyright to the text and the copyright term for them will be based on the lifetime of the artist. Likewise, if the book is a translation, it is the lifetime of the translator which determines the copyright term not the original author, unless you want to quote the original language version.There is a special case where the illustrations are photographs, but since you didn't mention photographs I won't complicate matters by discussing that here. Come back to the forum if you would like to know more about how to deal with old photographs in books.
If you think that a book and its illustrations are probably still in copyright there are several options. The first is to write to the publishers (or more likely their successors) and ask for permission to use the parts you need. This usually won't cost anything. Or if that is too difficult because you can't track down either the author's details or a modern successor to the original publisher, it may be possible to get something called an orphan works licence
which would give you immunity from a copyright infringement claim should an heir to the author come forward and identify themself as the present day owner of the copyright.
As far as using the textual information in a particular book is concerned, if this is largely factual (for example, it covers historical facts, rather than a work of fiction) then you can freely re-use this information provided you put into your own words and don't just copy large chunks of the text verbatim. You can quote short passages as long as you provide adequate citations for the author and the title of the work. Indeed this is fairly standard practice in academic research, and is made legal by section 30
of the Copyright Designs and Patents Act 1988*. Note that the description of this is 'fair dealing' and the emphasis is on the word fair. Don't overquote just to save yourself from having to re-write something on your own words. However, if you feel that your book would sound more authoritative by quoting what some other author(s) have said on the subject, then this will usually meet the fair dealing criterion.
Unfortunately this exception may not be appropriate for illustrations. Each individual image is treated as an artistic work in its own right. Thus in order to 'quote' an illustration you would need to use the whole of the work and this would usually mean that it is no longer fair dealing. It's hard to give concrete rules about this but re-using a graph or fairly simple graphic (along with the appropriate citation) would probably be acceptable, but copying a detailed engraving like a portrait would not.
* There is another fair dealing exception (section 29 for the purposes of research and private study) but this is not available for something which is to be commercially published and so would not apply if your book is to be published; it would apply if you were just writing an academic paper or thesis with no commercial motive in mind.