Is it much cheaper to be booking on the day as to buying it few days in advance?
No. Tickets are the same price whenever and wherever* you buy them.
But... certain routes have
Advance tickets available, which can only be bought before travel, typically from 12 weeks before travel until 1800 the day before (sometimes up till midnight). These tickets can be cheaper than "walk up" tickets -
(Super) Off Peak and
Anytime - but are only available in limited quantities. When the cheapest Advance fares sell out, the next cheapest are sold and so on.
For your journey, the "walk up" ticket would be an Off Peak Day Return at £8.30 (£5.50 with a Railcard). There are some Advance singles available for certain trains for £3 (£2) each way this Saturday, but unless you're really, really strapped for cash I'd not recommend buying these for such a short journey.
Bear in mind that Advance tickets are non-refundable and only valid on the train you've chosen; you'd have to buy a new ticket if you missed that train or wished to get an earlier one. Also bear in mind that most ticket booking websites will automatically select Advance tickets if they're even 5p cheaper than the "walk up" fares, always make sure you check the equivalent Off Peak fare in view of the lack of flexibility Advance fares have.
*certain train companies offer discounts on Advance fares for their own trains when booked through their own websites. "Walk up" tickets are the same price everywhere.