Re-cut damaged or worn lawn edges, cut back into good grass to form a clean new edge. If the edges are in poor condition it is better to replace them with new turf. Where border plants tumble over the lawn you can try to put a row of bricks round the edges as a cutting strip for the plants to spread over and where the plants are invasive just cut them back.
In mid spring, as time goes warmer you can start a weekly mowing regime but keep the blades set quite high at the start and then lowering it a little at a time as grass growth increases over the weeks. You can help it by applying a high-nitrogen lawn feed.
In mid spring you can start re-seed the bare patches from the lawn. Scratch over the are you want to re-seed with a rake, sprinkle with seed, cover with a thin layer of potting compost an water well.
Continue dealing with the moss problem just like in early spring and if any hard weeds are causing you problems, like daisies, dandelions and buttercups, just dig them out individually by hand.
As time goes warmer, in late spring, you can gradually lower the height of the blades on the mower as grass growth gets stronger. Start the cutting this season at about 2.5 cm and lower the blades in time as low as 1.25 cm for fine-quality lawn.
Keep applying fertilizer to your lawn, using quick-acting liquid fertilizer rich in nitrogen or slow-release granular fertilizer. Keep dealing with moss and weeds problems like you did before and re-seed all the areas where grass is very thin.