Saving tips for a wedding would be similar as for anything else, so 'pay yourself first' i.e. do a really good, realistic budget, work out how much you (and your fiance!) can afford to set aside for the wedding each month, then set up a standing order for the day you get paid so it barely touches your current account, no risk of it accidentally being frittered away. For me, I wouldn't ruin my life to pay for an expensive wedding so no beans on toast for dinner etc. but you could look at simple areas to cut back on non essentials such as under-used subscriptions or gym or other club memberships, next time your phone is due an upgrade keep your old one and go onto a sim-only plan, maybe do a few meat free meals a week to save on food costs (good for the planet too!), walk to places rather than using the car, if you are crafty make things as gifts for people for Christmas, birthdays and so on rather than spending lots of money? If you are in the kind of job where you can pick up some overtime easily or even get a second evening/weekend job that can massively help too!
Re the costs of the wedding itself, yes venue will likely be the no 1 cost, to save money depending on where you are you could look at a cheaper reception venue like a village hall, private room in a pub, or in London the site 'Tag Venue' has a great list of 'alternative' reception venues - of course most of these places aren't licensed venues for weddings in England (different in Scotland) so you'll need to consider either doing ceremony somewhere separate than transport to the reception, or do what some friends of mine did, have a celebrant (can even be a friend or family member) do a 'non official' ceremony but with vows and such, then pop down to a registry office to get legally married at another time which can be lovely. But the great advantage of a lot of the alternative venues is they may allow/encourage outside caterers and 'bring your own booze' (sometimes with a corkage fee) which usually works out cheaper for better quality than using the in-house caterers at a hotel or golf club or more traditional venue. You can then explore other options than the traditional sit down 3 course wedding breakfast which tends to work out expensive, I've been to weddings that did lovely buffets, hogs roasts, pizza, fish and chip vans, all went down very well! You used to be able to do a 'booze cruise' over in a hired van to France and pick up excellent quality cheap wine which worked out much cheaper in those days even with the van hire, ferry and petrol costs but do check how that stacks up these days plus customs rules since Brexit. It might be better just to go to CostCo 
I do think even at a budget wedding you need to consider the comfort of your guests and only invite as many people as you can comfortably feed and water well, if that means doing something smaller and more intimate with immediate family and very close friends only then so be it.
Obviously also massively de-prioritise things which won't particularly enhance your or your guest's days but are 'traditional' like chair covers, favours etc. If you have skilled friends and family you could ask for their help in lieu of a present - e.g. a friend with a decent camera instead of a pro photographer or videographer, musical people to perform instead of a string quartet/band, someone who can arrange a good playlist instead of a DJ (so long as venue has a good sound system or someone can bring one, home speakers won't usually cut it!), homemade cake or buying flowers wholesale and arranging yourself can save you £££ but do check yourself for bride-zilla-ish-ness, I have had friends that have expected professional-quality work from well-intentioned but obviously only amateur friends with things like this (and also the bride sometimes hasn't paid anything, even to cover costs e.g. of ingredients which can leave the kind friend very out of pocket!) and it's turned very sour which is a pity. If you are expecting perfection obviously you do need to pay!