In Italy, public transport is relatively cheap, efficient and convenient... You can ride the buses to all sorts of little places along the Amalfi coast, for days out. Wherever you are, look for 'spiaggia libera' to get free access to the beaches (they won't necessarily be as clean and well groomed as those where there are 'lidi' with a bagnino to keep it up to scratch. But the local authorities are supposed to keep them 'decent'.
In Sorrento, the 'resort' is on a cliff, and access to the sea is via lifts/steps and there is not much of a beach (black volcanic sand).... only bathing platforms off the cliffs into deep water. Charges are obviously going to be expensive, as someone has to pay for building the platforms ! But if you catch a bus down the coast, there are pebble beaches at Amalfi, which means the water is clear, and you can walk onto the pebble beaches without steps and lifts, with no swirly sand to make it look murky !
For food, you can buy filled rolls (panini) at a 'salumeria' (cheese and cooked/cured meats) and head for the early morning fruit and vegetable markets (mercato) for cheap fruit - they usually sell by the 'chilo' and 'mezzo chilo' (2lbs and 1lb) but you can ask the seller to do you a price for ....say 2 peaches, 4 apricots and a handful of cherries..... You don't want to store chilos of fruit in your hotel room. Although, if there's a mini bar, you could store some food in there. Take the bottles out but leave them in the room, and put them back in when you don't need the fridge. But leave no crumbs or bits around, keep the room pristine clean, as ants will appear at the slightest inkling there's food around (they'll march up the outside walls and in through your balcony ! )
The 'trattoria' is usually a more 'homely' type of restaurant (unless they're calling it that as a style thing).... and the sort of take-away places are : 'rosticceria ' and Tavola Calda... some 'tavola calda' places have 'stand - up ' or bar stool type seating. Pizze are obviously a good choice, as compared to prices in the UK, the food is plentiful and the pizze authentic (choose a place with a wood oven for the best type of authentic pizza and flavour).... don't choose a place that uses an electric oven the flavour is all wrong.
Check prices on restaurant menus, you may find that a lunchtime meal is cheaper and just as satisfying as an evening meal. So you can occasionally switch round, and have your 'picnic' in the evening, under the stars. Look out for the places where the local 'workers' go for lunch - they don't all go home ! Sometimes it's enough just to head away from the main tourist areas to find the more reasonable prices.
Head out in the morning for wherever you wish to visit, and look for somewhere suitable to eat at midday - if it's all a bit expensive, then opt for the picnic - find the salumeria / forno for your rolls (ask an Italian housewife !) but don't leave it too late as the non-tourist shops then shut till around 4pm, and perhaps an ice-cream for dessert.
Then find a trattoria for the evening.or opt for a pizza, eat-in or take -away according to your pocket.
Do searches on this TA site for this area (Campania) to research the value of various tourist board offers such as Campania Arte Card and the like, to keep entry fees down and travel cheap. Read up the various 'threads' as there's always information to gain from them, even if they don't appear to fit your exact requirements. There's always a supplementary question asked by someone on the thread, other than the first poster that gets the title up.
Go and listen to Italian music..... take advantage of the open-air lfe out there... just sit around and 'people watch' ... and the churches are free, and full of art / architecture to admire....free spectacles when they have processions and festivals in honour of a saint with fireworks and music (check the saint's days) and weddings and 'First Communions/Confirmations' are also 'spectacles' when everyone 'dresses up'...apart from their religious content ..... and the churches are cool on a hot day !
It's your honeymoon, so make it fun and romantic for each other.... you'll have a roof over your heads, so be inventive with food and entertainment. Enjoy yourselves...