Improving basic maths & algebraic manipulation skills I am currently doing a-levels in the UK (received an A in AS) so doing calculus (differential equations/trig integrals)/linear algebra/mechanics/statistics etc.

However I feel that a lot of my basic mathematics skills are almost non-existent (don't know how I made it this far). 

Can anyone recommend any good books that provide examples and exercises to help me improve this vital skill (or set of skills)?

Thanks
It is not an easy question to answer as everyone is different and learns differently and it is not clear if you want the theory versus applied path.

However, here are some suggestions for your consideration.

First and foremost, read, study, do problems, think about theorems from various perspectives, question, explore, practice, practice and then practice!

Many questions along these lines have been asked and you should certainly review these.

MSE


how to be good at proving? (see proof books in my response)
Filling the gap in knowledge of algebra
Self-teaching myself math from pre-calc and beyond.
Self-Studying Abstract Algebra; Aye or Nay?
Motivation and methods for self-study
Complete course of self-study
How to self study Linear Algebra
Rigorous Text in Multivariable Calculus and Linear Algebra
A rigorous book on a First Course in linear algebra
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Web Resources


Look to the web for notes.
Look at the curriculum at various colleges and look at topical areas, syllabus, books, homeworks, et. al.
Khan Academy
Open Courseware Consortium (for example MIT)


Additional Items


Learn some programming using a Computer Algebra System
I recommend Frank Ayres First Year College Mathematics (Schaum's Outline).  Get one of the old editions (hard copy, not a scan).  Quite cheap on Amazon.  

This book covers all of basic algebra, trig, analytic geometry, and has precalculus.  Since you have been exposed to the material before (but didn't learn it well), this sort of review book is perfect.  Lots of problems.  Text is very clear, simple and short.  Good examples.  And answers to all the exercises.  Just drill through that and you will solve your issue.

I was disagree with and am perplexed with several people recommending books on proof or the like.  Non responsive to what the person asked for.