I live in a rented house with a garden in North East of England. Last year I attempted to create a small bed for growing herbs and some vegetables but the soil appears to be very clayey and wet all the time so my there was no harvest last year to speak of. After reading through other questions here I have decided to buy some compost and mix it into the soil. Should I also mix something to improve the drainage of the soil? I have found advice suggesting using grit, sand or perlite. Which one of them is the most suitable? Also is there something else I should do to improve my harvest? Please keep in mind that I do not want to invest large amounts of money into it.
Apart from the heavy soil, your primary problem is lack of sunlight in the growing area. What you need is somewhere that gets all the sun that's going, but a minimum of 8 hours a day is essential (during summer). Herbs and vegetables all like full sunlight, though they will cope with a little shade, and a greater amount of sun exposure would reduce the wetness of the ground. In respect of perlite, that's not something you'd put in the ground, its normally used in pots, and sand does very little to improve clay soil.
As you don't want to grow root crops such as carrots, then digging the soil over thoroughly to a spade's depth, incorporating a large bag or two of horticultural grit plus plenty of soil conditioning compost such as composted manure will help to 'lighten' the soil. You will need to do this over a wider area than you actually want to use for planting; if you just do a small area and that is still surrounded by unamended, undug soil, any water there is will drain straight into your newly dug area. Even so, you've still got the problem of not enough sunlight.
Raised beds placed on top of the soil, filled with good, screened and high loam content topsoil and some composted material would be another way forward, particularly if raising the growing area up would mean more sun exposure. It's likely the raised beds and soil may cost slightly more than thoroughly preparing and emending as described at soil level, but you can buy raised beds or even troughs on legs, preferably with an internal depth of 8 inches or more, rather than 6 inches or less. These will need watering during summer, but you could take them with you when you move for use elsewhere, so you won't lose to much of your investment, provided you keep the wood they're usually constructed from in good condition ongoing. It's certainly worth considering this latter option if it definitely means more sun exposure.
The other possibility is to grow in large containers or pots,which again will need regular watering during summer; the courgettes and other squashes might not be so easy in pots because they tend to sprawl far and wide so would need tying in to upright supports, but the other things you've mentioned will do fine in containers, although again, with the caveat of a lot more sun exposure being essential.
You've asked what soil to use if you build a raised bed; if you mean a raised bed sitting on the existing soil, open at the bottom onto the soil, then you can use some of the soil from your garden, but just mixing in some multi purpose potting compost with it won't achieve much - you'd more or less have the same growing conditions you have now, only with a reduced soil level somewhere else in the garden, and a raised area with more or less the same soil as is already there inside it.
If there's somewhere in the garden that you can remove a layer of topsoil that would half fill your raised bed without leaving a noticeable dropped area, then do that, and make up the rest using bagged topsoil, as well as adding in some composted materials, mixing it together well. Pre bagged topsoil is available at the garden centre, but you will need a fair number of them to make up the soil level in your raised area - they are usually no heavier than 40 litres or thereabouts for ease of lifting and transporting. Otherwise you can have topsoil delivered in much larger bags separately - if you choose that option, make sure you select graded, screened, high loam topsoil.
I think an explanation regarding the term 'compost' is essential here - in the UK, we use the term 'compost' as a sort of catch all term; what's important are the words in front of that word 'compost'. You can get multi purpose potting compost, John Innes potting compost, seed and cutting compost and other potting composts intended for use in containers -in other countries, these are more sensibly referred to as potting soils rather than potting composts. They have been sterilised so that they are pathogen free.
Separate from those are soil conditioning composts which have not been sterilised and are humus rich; they will include composted animal manure, composted horse manure, and a range of other composted materials in bags, depending on your local area. In some areas, you may be able to get hold of composted leaf mould or spent mushroom compost - all of these are intended to be used on open ground, but not intended for use in containers (because they are not sterilised and may contain pathogens which are fine in open ground, but not fine contained in pots). Your own garden compost from a cold, anaerobic heap or bin falls into this category of soil conditioning material. Pre-used, spent potting soils also fall into this category, for they will no longer be sterile, though they will contribute little to the humus content of your soil.
It is this latter type of soil conditioning compost you want to mix into your raised beds, and I reiterate, this assumes your raised beds are completely open at the bottom onto your existing garden soil. Note there is no negative effect from using potting compost instead of soil conditioning composts to mix in, but they are more expensive and far less effective at providing humus rich materials; raising the humus level in soil is the most effective way of keeping the bio diversity of the soil up so it is in top notch condition to grow your plants, as well as improving both heavy clay and light, sandy soils. I'd recommend using composted animal manure, whether that's horse or other animal, but don't rely on this to 'fill up' the level in your raised bed - composts like this shrink down over time (because the humus material is being converted and used by living organisms in the soil) and won't increase the soil level by much, only topsoil will.