Just a decent mower, preferably one that collects the cuttings, is all that is needed. The 'floating' Flymo type may not 'float' as well as adverts claim, so a wheeled mower may be better. 'Cylinder' mowers produce the smoothest cut, if they are accurately adjusted, but rotary ones are OK for a normal domestic lawn.
As you surmise, don't mow if grass is too wet, and don't cut it too short yet, as it is still early in the season. Some people mow two or even three times a week, so it will look untidy if left for two weeks. If it's an electric mower with a cable, have a RCD cut-out (residual current device) [Masterplug £9] so if the cable does get cut, you don't get electrocuted!
I don't like strimmers, and if you have much edging to do long-handled edging shears make a neater job. If you have areas of rougher, long grass I guess strimmer is easiest, or ordinary shears, and down on your knees on a mat if you can manage that. (We have Wolf shears, and they are the sharpest, easiest to use I've ever had.)
If you want to get 'thatch' (dead grass) and/or moss out, a lawn rake is cheapest, but hard work; Wolf do a wheeled scarifier that fits on their standard long handle, and is easy to use. You can also 'aerate' with a fork, or a hollow tine aerator, but only the keenest lawn enthusiasts go to those extremes.
But regular mowing is the most important thing.
Good luck, and come back if you need any more help.