How did that work

It was up to the legislature how to set that up. In the first federal elections in 1788-89, some legislatures did it by joint resolution of the two houses (e.g. Maryland, Delaware, and later North Carolina once they ratified the Constitution), and others did it with each house electing and then the other house concurring, sort of like what happens with bills (e.g. New Hampshire, Connecticut). Some states only had one legislative chamber (e.g. Pennsylvania, Georgia) so they just held a simple vote. A state could set up special rules for itself if it wanted to. For example, Maryland's legislature in that first election in December 1788 specified that one senator had to come from the Eastern Shore and the other from the Western Shore (the east and west sides of Chesapeake Bay).

It would be a lot of work to give a comprehensive answer for all the states for all of 1788-1913. You'd have to be familiar with the legislative histories of every state. The federal government left it up to the state legislatures to decide their own procedures.

Here's an example of the election of a senator from Utah in 1903: https://newspapers.lib.utah.edu/details?id=2435024. Scroll down to "Special Order" and "Result of the Vote".

and how were State Senators chosen?

They were elected by the eligible voters. That was the typical procedure, and I'm fairly confident that was the universal practice by all states since the beginning. Normally the state was divided into senatorial districts, and the voters in each district would elect one or more state senators. There were exceptions to that. I believe in Connecticut state senators (of which there were originally twelve) were for a time elected at large, though that is not the case today. In some states, state senators represented counties (e.g. New Jersey's 1844 constitution). That practice was ruled unconstitutional by the U.S. Supreme Court in the 1960s (a little ironic, given how the federal Senate works), and now state senators always represent specially-drawn districts with reasonably equal populations.

When writing the above paragraph I went to https://en.wikisource.org/wiki/Portal:Constitutional_documents#United_States and skimmed a few constitutions just to make sure.