Plant varieties for sowing and planting are fairly similar for both June and July in most climate zones of Australia. Peas and broad beans should be sown at a time that will avoid frost damaging the flowers see: Broad beans and peas)
Sowing times are more flexible when sowing peas or broad beans as a green manure because these crops are cut down as soon as the plants start to flower.
The following planting guide is an abbreviated list for vegetables, fruit trees and some culinary herbs that can be planted in June in Australia and New Zealand. A comprehensive monthly guide that includes planting times for the entire garden, as well as when to fertilise, prune, weed, take cuttings or divide plants, can be found in my book Easy Organic Gardening and Moon Planting.

WARM CLIMATE – South of Rockhampton
Before the Full Moon, cabbage, and grains can be sown directly into beds, as well as a green manure crop of barley, chickpea, red clover, broad bean (faba bean), field pea, or triticale. Lettuce, radicchio, English spinach and spring onions can be sown or planted out.
During First Quarter phase, dwarf peas can be sown directly into beds.
During Full Moon phase, radish and turnip can be sown directly into beds, as well as potatoes north of Brisbane. Asparagus and rhubarb crowns, fig, kiwi fruit, pecan and pistachio can be planted.

WARM CLIMATE – Rockhampton northwards
Before the Full Moon, cabbage, open Chinese cabbage, grains, lettuce, mizuna, rocket, silver beet, tatsoi, chamomile and coriander can be sown directly into beds, as well as a green manure crop of barley, corn, lablab, or triticale.
During First Quarter phase, bush and climbing beans, popcorn and sweet corn can be sown directly into beds, and pumpkin, spring onion, summer squash, tomato, watermelon and zucchini can be sown or planted out.
During Full Moon phase, beetroot, radish, turnip can be sown directly into beds, and fig and pistachio can be planted.

TEMPERATE CLIMATE
Before the Full Moon, English spinach can be sown directly into beds, as well as a green manure crop of broad bean (faba bean) or field pea. In frost-free areas, lettuce and spring onions can also be sown or planted out.
During First Quarter phase, broad beans and peas can be sown directly into beds.
During Full Moon phase, garlic and radish can be sown directly into beds, and mid season onion seedlings, asparagus and rhubarb crowns, kiwifruit and pistachio can be planted. In frost-free areas, fig can be planted.

COOL CLIMATE
Planting is extremely limited during both June and July. Before the Full Moon, English spinach can be sown directly into beds, as well as a green manure crop of broad bean (faba bean) or field pea. Broad beans and peas grown as a vegetable can be sown during First Quarter phase (see link to the post on when to sow Broad beans and peas for your local climate).
During Full Moon phase, mid and late season onions can be sown, and asparagus and rhubarb crowns can be planted, also deciduous fruit trees and vines where frosts are not severe.