You have the basics set down, but there is always more to it.

Firstly the February Revolution was achieved quite easily. Once the army discipline broke down in Petrograd, it began to spread everywhere. Instead of a bloody coup happening, the Tsar was convinced to abdicate. Tsar Nicholas's son was quite sickly and Nicholas being the family man abdicated his son as well in favour of Grand Duke Michael. However Michael refused the throne. Russia unexpectedly found itself as a republic.

These go wrong off the bat. Two governments are set up. The Petrograd Soviet (PS) and the Provisional Government (PG). A soviet is essentially a government body made up of the proletariat/supporters. So Mensheviks, Bolsheviks, etc. The PG was another body set up by Duma members. It was initially unelected and derived power from the Duma. The two bodies agree to share power until a constitution is made. It can be shown that both bodies were competing especially when the Bolsheviks gain traction.

The PG had many problems, an coup attempt initially wasn't one of them. Many Tsarist leaders supported the PG. Despite this many of them were fired due to Tsarist links. The PG had poor governing and administrating skills, in many cases they didn't replace tsarist governors. The vacant seats were left unfilled or were assumed by local soviets. The railway system began to break down; the administration was inefficient; laws were ill thought out; army discipline began to break down especially after the Kerensky offensive; shortages in the cities. These are just some of the other problems that meant that the PG never got true support from the peasants and proletariat. That all went to the PS.

So why did it take so long? Well the problems didn't become acute until later on, really only after July Days protest did the problem reveal itself. Additionally the PS didn't want to take power, rather the Bolsheviks were the only meaningful party who wanted too. They had to wait til they got enough support before they launched the coup. This was only in November (on our calender).

I suggest reading Sheila Fitzpatrick's book: The Russian Revolution. It's not too long but covers this period in a good amount of detail. It's a great starter, then you can delve deeper.