Whether you are researching plants to satisfy your own curiosity, or because you are thinking about registering a plant in the Plant Guardian scheme or having a National Plant Collection, there are a lot of resources to help you on the internet.
If you don’t have free access to a computer, I suggest that you enquire about using one at your local library, or share with a friend.
Plant names and taxonomy change over time; to get the latest information on plant families start here: (http://www.mobot.org/MOBOT/research/APweb/) with the Angiosperm Phylogeny Group website. (No one person or organisation is ever 100% in agreement with other people or organisations on plant names. Find a source that is scientifically reputable and accepted by as many other people as possible!)
If you think that the area you are interested in is currently being researched, try Google Scholar (https://scholar.google.co.uk/) which will search academic papers for you (bear in mind that a lot of these are not free to access).
The next place is the Plant List (http://theplantlist.org ) where you will find names and synonyms of plants of wild origin. For cultivars in the UK, the best reference is the RHS Horticultural database: (http://apps.rhs.org.uk/horticulturaldatabase/HortGenera.asp), although individual species or genera may have better references via a cultivar register.
Not all plants will have a cultivar register, it tends to depend on the enthusiasm and organisation of growers, many are listed here, though they are not all great at communicating (http://www.ishs.org/sci/icralist/icralist.htm ).
To find out if your plant is grown widely in the UK, check to see how many sellers it has the RHS Plant Finder (access via the RHS Horticultural database), Google (though do take nursery lists with a pinch of salt!), and garden databases. You can check the RHS gardens, Sir Harold Hillier Arboretum & Royal Botanic Gardens Edinburgh (among many overseas gardens) using this site: http://rbg-web2.rbge.org.uk/multisite/multisite3.php
You can search Kew & Wakehurst Living Collections via http://epic.kew.org ; and a wealth of other information here: https://www.kew.org/science/data-and-resources .
If you are more interested in the history of a plant, start with the Biodiversity Heritage Library: http://www.biodiversitylibrary.org/ which is a consortium of natural history and botanical libraries, offering a mind-boggling range of books, prints and catalogues online.
If you would like to find herbarium specimens online try https://plants.jstor.org/ ; or http://linnean-online.org/linnaeus.html, http://data.nhm.ac.uk/ (the last two have many more options than plants!).
And for historical research based on a garden rather than a plant try here: http://www.parksandgardens.org/ .
Websites change very rapidly, so this information is likely to be out of date within six months or so. If you can’t find one of the resources listed, put the name of the resource you want (e.g. Kew Living Collections Database) into a Google search and you will probably be able to find it.
Links correct in Jan2018.