A lightly edited version of a previous answer:

Firstly, remember that one primary reason for the colony - perhaps not the only one - is that the UK upper classes really did want to get rid of the criminal element in their society, without necessarily just executing them. To quote from Alan Frost's Botany Bay: The Real Story,

By the beginning of the eighteenth century, the view was widespread that the law was defective because it offered only a few punishments intermediate between death and branding or whipping. It was to remedy this defect that in 1718 parliament enacted legislation providing for felons whose crimes had not been of the most extreme nature to be transported to Britain's colonies in America for terms of seven years, fourteen years, or life.

This wasn’t the only means of dealing with ‘Intermediate’ criminals that began in the 18th century in Britain: the 'house of correction' developed, where people spent time forcibly working 'hard labour' for periods of three to twelve months (the ancestor of the modern prison). But transportation remained the preferred form of dealing with 'intermediate' criminals until 1776 and all that, when Britain no longer had control over the colonies it had controlled in North America.

In the intermediate period after 1776, the prisoners who would have been transported to America were instead stored on ships moored on the Thames. As the ships became increasingly crowded, and as it became increasingly clear that America was not going to be easily reconquered, the British started to consider where else in the world where they could send their convicts. In a series of parliamentary inquiries through 1782-1784, the British debated where to resume transportation. By December 1784, the British had three options that they considered seriously, which were all basically either outside the then-British Empire or in thinly-established areas of the empire (a clue as to part of what they were going for):

1. Transportation to the pre-existing (slave trader dominated) Cape Coast Castle on the 'Gold Coast' of Africa (now Ghana)

2. A settlement in Lemain, up the River Gambia, in Gambia; and

3. A settlement in New South Wales, an area of the world had recently been visited on one of Captain Cook's expeditions, and which was pushed by members of Cook's expedition, like James Matra and Joseph Banks.

In August 1783, James Matra suggested to the Home Secretary that Botany Bay had a climate and soil which:

are so happily adapted to produce every various and valuable production of Europe and of both the Indias, that with good management, and a few settlers, in twenty or thirty years they might cause a revolution in the whole system of European commerce, and secure to England a monopoly of some part of it and a very large share in the whole.

Matra continued with the observation that there were strategic advantages to Sydney:

The place that New South Wales holds on our globe might give it a very commanding influence in the policy of Europe. If a colony from Britain was established in that large tract of country, and if we were at war with Holland or Spain, we might very powerfully annoy either state from our new settlement. We might with a safe and expeditious voyage, make naval incursions on Java and the other Dutch settlements, and we might with equal facility invade the coasts of Spanish America, and intercept the Manila ships, laden with the treasures of the West.

Initially, the UK favoured the African solutions; indeed, 13 convicts were sent to Cape Coast Castle in 1783-1784 as a trial, and there was a proposal to send 200 convicts to Cape Coast Castle per year, but the head of the British presence at Cape Coast Castle found the 13 convicts a burden and did not want more.

Instead, on December 27th, 1784, the decision was made by the Lord Sydney and the British Cabinet to send convicts to purchase the island of Lemain in Gambia from the natives in order to establish a convict colony in West Africa. There were dissenters from this decision; one Edward Thompson warned Cabinet that the British public would see it as essentially sending convicts to their certain deaths, because of how dangerous Africa was, and that the Opposition would make hay.

These plans leaked in March 1785, and Edmund Burke indeed said in sorrowful tones from the Opposition benches that it was sending convicts to a worse fate than a quick, just English death. Parliamentary committees heard testimony from doctors that Africa tended to result in 'putrid fevers and fluxes' in Europeans without much medical attention. The government's plan changed from Lemain to Das Voltas Bay (now Luderitz Bay) in Namibia, but the Opposition pushed harder for the Australia colony through 1785 discussions. The British sent a ship, the Nautilus, to investigate Das Voltas Bay, and in the time in between the Nautilus expedition setting off and returning in July 1786, the political winds changed. Ultimately, the Nautilus expedition reported that Das Voltas Bay was unfavourable grounds for a colony, because it did not contain much of an inlet (and so no fresh water supply for the colony).

Ultimately, a settlement in New South Wales won the day. Frost lists seven advantages of a New South Wales colony that were discussed in 1786:

1. Botany Bay was able to offer a 'sheltered anchorage', and one able to be fortified

2. The environs of Botany Bay had the potential to meet the needs of the colony, according to the opinion of Banks, with a number of streams, plentiful fish and birdlife to help support feeding the colony, with timbered lands to supply wood, and lands that might be suitable grounds to raise cattle or sheep or to plant crops (for better or worse, this was not what was found at Botany Bay in 1788 - it's likely that when Cook arrived in 1770, Sydney was in a La Nina period, and in 1788 it was in a El Nino period, with some severe climate differences as a result).

3. The climate of Botany Bay appeared 'healthy'/suitable for Europeans (unlike the 'putrid fevers' of Africa).

4. The British could claim New South Wales without offending any of their European neighbours (unlike the African colonies, which might be seen as threats by other European colonisers).

5. The British in 1786 incorrectly believed (on the words of Banks and Matra) that New South Wales was so thinly populated by people who were not naturally owners of the land that New South Wales was effectively what we'd now call terra nullius - unlike in Lemain, the British wouldn't need to purchase the land of the colony from the natives.

6. Sydney was conveniently located, strategically (see Matra's points above about its location vs Java, etc) allowing the British to quickly attack the Spanish in Manila or the Dutch in Java, or to trade with South America provide support to British interests in India; and finally

7. The Pacific offered an abundance of useful natural resources; e.g., it was always the plan to have a secondary settlement on Norfolk Island, due to its pines, which were excellent for shipbuilding and ship-repairing.

As to what the UK hoped to get out of transporting convicts across the world, opinion amongst historians differs. For Robert Hughes in The Fatal Shore, the British authorities generally thought that there was a 'criminal class' amongst England who they barely understood (living in very different upper class worlds), but who were incorrigible, and who they simply wanted to get rid of and forget about. Grace Karskens in The Colony argues that Sydney was specifically founded as a rural colony, which was meant to avoid the corrupting influence of the city that some of the British authorities believed played a role in corrupting the criminal class; if you transported them away from that situation and into a situation where they'd learn the value of honest hard work, living off the land, it'd reform them. In either case, the convicts also provided a cheap labour force for building a new settlement to achieve British policy objectives, and a labour force that wouldn't really be missed if the colony wasn't a success and they all starved (as it did come close to at a couple of points).

So, broadly speaking, there were a bunch of (sometimes conflicting) policy objectives that the Sydney colony was meant to accomplish, and which weren't always prioritised by the governors who ruled the colony, who had to deal with the reality of conditions in the colony. Governor Arthur Phillip seems to have struggled to obtain and find appropriate convicts to stock the colony with - very few of the people who landed on the First Fleet knew very much about farming, for example, and part of why Karskens argues that Sydney was meant as a predominantly rural colony was because no real plans were made for Sydney as a port city in the early years of the colony - the authorities in Britain seemed to not want to approve such things. If you see the policy objectives of Sydney as listed by Frost above, it's a bit odd that Sydney wasn't quickly established as a port city, but this just speaks to the varying motives of different parties involved in establishing the colony.

As to why Western Australia (or anywhere in Australia that wasn't Botany Bay) wasn't on the menu - well, Captain Cook hadn't sailed around the west coast of Australia on that voyage in 1770. He'd sailed across the Pacific, made it to Tahiti, investigated New Zealand, and then sailed west to Australia, making sight of land in Southern NSW, before sailing up to Queensland and then onto Jakarta. By the 1780s, Banks and Matra (who'd been on that expedition) were respected enough to be discussing Australian options with decision makers, and they felt that Botany Bay was the best Australian option; they had personal experience of Sydney. Perth, e.g., was never discussed.