Firstly, use Firefox web browser and install the 'Property Bee' plugin (Google it). It inserts a history box into listings on Rightmove to show how long they've been on for, price changes, sales fallen through etc. You can quiz agents with queries you already know the answer to to weed out any dishonesty etc.
Secondly, check the last sold price on Zoopla (goes as far as 1995).
Also, check out last sold prices of similar places on Zoopla or houseprices.co.uk. Fair value may be somewhat different to the asking price, so don't be embarrassed to put in a low offer. The agent may attempt to get you into a bidding war, and the other bid may or may not be real depending on the scruples of the agent, so don't be afraid to say you are not going any higher and the other party can have it.
Those three things ought to put you in a decent position, and I bet most potential buyers don't bother doing any of it.
I would also make sure you are very clear with yourselves about what you can afford and stick to it. That also means (in the case of leasehold properties, especially flats) factoring in any ground rent and/or service charges (which can be hefty). Get as much information on this as soon as possible if it's a place you are seriously interested in, preferably prior to commiting to spending money on a survey etc.
If you are the sort of person that 'falls in love' with a particular property, then that may costs you dearly. If it is the case, try and put it to one side. There are shedloads of houses around, don't get fixated on one.
Finally, please remember that the agent works soley for the seller. Their duty is to get as high a price as possible for the seller. They couldn't care less if you end up loading yourself with an unsustainable debt mountain, so don't expect them to do anything but encourage you to spend as much as possible.
Finally +1, if it smells bad, walk away. If you feel uneasy about info supplied or any aspect of a property you are interested in (maybe it's had three sales fall through on Rightmove, or the agent is cagey about coming up with some info), just trust your instincts and walk away.
Good luck.