Ist thing to do is to get some Dr Hessayon "Expert" books. Usually found on a rack in garden ctrs. They are cheap paperbacks which are very informative - one on lawns , on shrubs ,flowers ,bulbs etc. Very informative.
2) Clay soils?

Thye go harde in the dry weather and heavy when wet. Drains poorly as well so you will need to work in a lot of organic matter to improve soil structure- peat compost etc, The mnoore the better. you can also lay some dowmn as a mulch round the plants and let the earthworms( gardebners friends) take it down I would also look to mix in some sharp sand to improve soil texture. Your plants will thank you for all this.
3) Bear in mind that its horses for courses , different plants thrive in different locations so get to know hteiur requirements. Some tolerate most shady conditions others like sunny dry areas which drain well.i always recommend Busy Lizzies Begonias and Geraniums as they give by far the best value and are relatively easy to grow. The first two tolerate shade as well.French marigolds are ok too but slugs love them so sprinle pellte round them onec a week( in this weather).That your flowers which are hte garden fillers
4)Have a framework of shrubs and climbers. You've got a small garden so dwarf shrubs like small hebes, euonymous, fuschias, potentillas. Climbers - ivy, clematis,Jasmine, honeysuckle. The last 3 will need training so attach some wires horizontally onto the fence or trellis.Eunuymous will also grow up the fence. Think in terms of heather for winter colour as well. I see you got a Cordyline which is good as it gives a focal point but bear in mind it is a tree and will gorw to abpout 10ft maybe more.
5)Lawns are high maintenance nad if dogs have been on it then

itwill take time and effort to get right but as long as you see improvement then it should encourage you further. You could scarify it with an electric rake to get rid of thatch,apply mosskiller if necessary, aerate with a fork to relieve compaction. There are special aereating tools with hollowed tynes which can be bought which would be better.Finally in autumn apply autumn feed and top dressing (can buy in bags)
biggest favour you can do yourself is improve soil condition as it will make weeding so much easier- clay is 'orrible to work on

AND BUY THEM BOOKS