P.S Is it better to go on a guided trip first to learn the ropes?
I doubt it: guides are a sensible option for things outside your experience where technicalities like rope work or skiing smarts can make the difference between a great time and death, but going for a walk with a camp really isn't that serious. The only difference between what you propose and what you're already doing is carrying a bit more and stopping overnight. Start in summer when good weather is forecast and you should be fine, plus an added buzz of doing it yourself.
With a budget of a grand you're spoiled for choice. You may well want a bigger (but not enormously bigger) pack, but again with your budget that's not a big issue.
I'd suggest something to cook with, or at least make a hot brew, and as well as a sleeping bag a good mat will make life much more comfortable.
Here's one possible breakdown of how I might spend that money to do what you're after. I've not shopped around for the best prices and this is just one set of many possibilities.
Alpkit Pipedream 600 sleeping bag £170
Alpkit Airo mat £40
MSR Hubba HP solo tent ~ £300
Primus Gravity EF stove ~ £60
Lightwave Fastpack rucksac k~ £100
On top of that you'll need various sundries like pots and pans, gas for the stove, but that gets you some very reasonable kit and leaves plenty of change.
For a rucksack it's very important you try them on with a representative load inside. Different folk come in different shapes and what works for one person may be torture on another. Similarly, if you can have a crawl around a few tents that's time and effort well spent, as on cold and rainy nights you'll be spending a lot of time looking at the inside...
The sleeping bag suggested there should do pretty much all year, but you can get away with lighter in the summer. Lighter means less to carry, and less to pay too.
For your first trip, make it the garden (or a pal's if you don't have one). Sounds pathetic but it means you can get familiar with layout, pitching and what does and doesn't work right next to an easy bail out.
Pete.