I mainly grow from seed but no reason why you can't grow from plug plants - larger garden centres will have a good selection. It does put the cost up, but then again if you grow from seed you need pots, labels, compost, trays. These can all be reused but there is an upfront cost. I have a greenhouse now but used to manage with one of those mini patio ones and working on the patio table. I have a long, thin windowsill incubator (plastic tray with a deep cover). I make tiny pots from newspaper, I have a gadget for this, and start any seeds that need warmth on my living room windowsill in it. Transfer to greenhouse. Others can be started in the greenhouse or directly into the ground, the packs have instructions.
Of those you list:
Onions - grow from sets (tiny baby onions) plant directly into ground any time now.
Lettuce - need to be frost free, start in incubator, transfer to greenhouse then ground, early in the year, can sow direct into the ground in summer but beware slugs.
Beans - incubator, greenhouse, plant out - frost sensitive.
Purple sprouting brocolli - incubator, greenhouse, plant out, grow slowly all winter, need netting.
Broad beans and peas can be sewn directly and early.
I would suggest going to a garden centre and reading seed packs to get an idea, also there are books and planners. Just being on this thread is very helpful as you can take your cue from what others are doing. One thing it is important to know is the predicted last frost of spring in your area, this can be quite localised, mine is late despite being in the SE of England. There are online frost maps.