This is a complex topic but I'll give it my best stab!

I thought the Iranian public had a very positive view of the U.S. at the time and this started off a chain of events that led the very hostile relationship we currently have

Firstly, you're correct that the Iranian public had a relatively positive view of the US but I'd be careful of suggesting that the coup "started off a chain of events that led the very hostile relationship we currently have". This can present an overly inevitable view of history; people are quick to link the coup to the Islamic Revolution but do bear in mind that they're more than 25 years apart and plenty could have gone differently in that time.

But to get to your main question, the UK and US both have several important motivations which different historians give different weight. I will separate them roughly into economy, strategy, ideology.

Economy

It is hard to overstate the value of the Anglo-Iranian Oil Company for Britain here; it is Britain's single largest overseas asset at a time where the country is trying to rebuild itself from World War 2. It is a vital source of dollars in a very literal sense given Britain's balance of payments woes at this time.

Beyond this, they were not the only player that stood to gain economically from the coup. The US had been frustrated by Britain's restrictive control on Iranian oil which denied US oil companies market access; following the coup this arrangement was clearly unsustainable and US companies were able to enter the market much to their benefit.

Strategy

As the British Empire was increasingly called into question, British strategists increasingly turned towards Africa and the Middle East as a solution to secure Britain's global position. The Anglo-Iranian Oil Company is not only a key resource: it's a key aspect of British strategic influence in the region (along with the Suez canal). For an easy example, just consider the importance of a secure oil supply through the major wars that had just passed.

Britain's strategic position is also a concern for the US. The US and the UK have just come out of WW2 where they fought as allies. They didn't always see perfectly eye to eye, but nonetheless they had an important strategic relationship. This especially true as the Cold War Era commences and the US is increasingly concerned with the spread of communism.

Ideology

The Iranian Oil Crisis is fascinating for the way which it highlights the balance between different ideological paradigms: imperialism, nationalism, and "cold-war"ism.

You've asked why the UK pursued the coup even though Iran was acting within its rights as an independent nation. This reflects a modern conception of nationalism -- and one which Iranian nationalists were quick to uphold -- but which wasn't necessarily that convincing to the British imperialist mindset. At least not when vital resources were on the line.

For the US, the ideological confrontation with the USSR -- and the possible spread of communism -- was a growing concern. Mosaddeq had wide popular support and something of a socialist platform. He also unfortunately also played to US fears of Iran falling to communism in an attempt to gain aid from the US; some later American sources further suggest that the British deliberately played on this fear to push the US into action. It's difficult to know exactly how much weight to give the fear of communism since naturally its a nicer justification for the Americans involved than oil-money. One thing that I would highlight here is the distinction between the Truman and Eisenhower administrations. Under Truman the US takes a generally reconciliatory approach with significant efforts towards a negotiated settlement. Eisenhower's administration (which is generally further into the paranoia of the Cold War) takes office and the coup follows shortly after.

Regarding why the UK asked the US for help, on top of the close economic and strategic relationship above there is also an important practical factor: the UK's ability to orchestrate a coup is hampered after Mosaddeq expelled Britain's diplomatic mission in 1952 and working with the CIA helps them to overcome this obstacle.

Closing thoughts

Firstly, I have separated various factors out, but please don't read them in isolation. For example, Iran's strategic importance should also be read in terms of the post-war geopolitcal orientation towards Cold-War competition between the US and USSR, and the American desire to open the Iranian oil market has an ideological undercurrent as well as an economic rationale.

Lastly, I do want to re-iterate that this is a really fascinating question which remains debated in the historical community. Over-emphasising the "fear of communism" interpretation risks giving too much weight to post-hoc explanations given by Americans and arguably verges on apologia. (This is also complicated by the fact that detailed American sources were more readily available than others). At the same time, over-emphasising the "it's all about money and power" interpretation risks boiling complex ideological and personal factors down to simplistic realpolitik. An interesting questions to ask yourself as you delve into the topic is "why does the US behave differently around the Suez crisis only three years later?".

Main sources:

Gasiorowski and Byrne (Ed.), Mohammad Mosaddeq and the 1953 Coup in Iran
Katouzian, Musaddiq and the Struggle for Power in Iran
Louis, The British Empire in the Middle East, 1945-1951
Bill and Louis (Ed.), Musaddiq, Iranian Nationalism, and Oil
Galpern, Money, Oil, and Empire in the Middle East: Sterling And Postwar Imperialism, 1944–1971

For more accessible reading, I recommend Gaziorowski, 'Coup d'etat of 1953' in the Encyclopaedia Iranica which is an incredible peer-reviewed online resource for Iranian history.