Battery recycling is commonplace - DO NOT PUT BATTERIES INTO YOUR WHEELIE BINS as they are a real fire risk when mixed in with paper and card (near us, 3 local bin wagons have caught fire due to batteries recently), or if they end up in landfill, dangerous chemicals leach out of them.
Retailers who sell more than 32kg of batteries each year are legally obliged to accept batteries for recycling in their outlets, this has been the case since 2009. There should be a collection container in store. To find your closest battery recycling point type 'batteries' into this Recyle Now webpage.
If you want to really reduce the number of batteries you get through, consider a gradual switch to rechargeable batteries, which can be found in most large supermarkets.
They are more expensive than regular batteries and you will need to buy a charging device to fit them. This Energizer Universal Battery Charger from Argos looks good as it can hold AA, AAA, C, D or 9V batteries, BUT as with virtually all such battery chargers you have to charge the batteries in pairs (to complete the circuit) which isn't helpful if you need to charge up an odd number of batteries. Sigh.
I would definitely avoid buying unbranded batteries and unbranded chargers, ideally buy them from a trusted 'bricks and mortar' retailer, not from an online marketplace as the risk of fake electrical goods is high and the consequences can be devastating.
There are different brands of rechargeable batteries to choose from - I'd always go for the big-names such as Energizer, Duracell etc, we always buy Uniross, as they have great milliamp hours (mAh) capacity values - the amount of electrical charge stored inside the battery (the higher the number, the longer charge can be delivered to your battery powered device). AA batteries will have a higher mAh value than AAA batteries. Look for AA batteries with 2500 mAh and AAA batteries with 1000 mAh (Uniross rechargable batteries offer these values).
There are different types of battery, e.g. NiMH and the currently less common NiZn, but old NiCad are no longer sold and don't always comply with modern chargers - so do check before you put old batteries into a new charger!
Some rechargeable batteries now come ready-charged, meaning that you can use them straight away (they always used to need a charge first, which would put people off, so this development feels like good news to me).
This Which? article has lots of useful information about rechargeable batteries (but you would need to be a subscriber to see the comparison tables at the end).
When your rechargeable batteries finally do give up, you can recycle them at the usual battery recycling collection points.