Do I likely speak German and Hebrew and Yiddish? If I meet a fellow Jew from Warsaw or Kiev, would we have to fall back on Hebrew or do we likely speak the "same" Yiddish? Was Hebrew reserved solely for study and worship, or would I likely live a tri-lingual existance?
This is a pretty good question!
At this point in history the difference between German and Yiddish was much smaller than today. Jews were using the Hebrew alphabet to write their vernacular language, but this is really at the very beginning of having written evidence of it. While they would've mixed other languages' vocabulary into their vernacular speech (esp for religious terms) in ways their non-Jewish neighbors wouldn't've, they probably wouldn't have thought of it as a distinct language. So you'd probably speak German, of if you like Judeo-German, and this would've been the same language you would've spoken to your neighbors in.
1250 was the very beginning of large-scale Jewish population in Poland. Since there would've been very few intervening generations for the Yiddishes to diverge, it's quite likely that they would've been able to converse without too much trouble.
Kiev is a bit more interesting. The Jewish community in Kiev was established well before 1250. However, unlike the population in Poland they weren't descended from German Jews who moved east. They probably were a mix of Jews who migrated to the region earlier and maybe also Khazars (who would've spoken a Turkic language). The descendants of these Jews survived long after in Crimea, known as the Krymchaks. So, definitely not Yiddish speakers. Krymchaks and Khazars would've originally spoken a Turkic language, though it's possible that by 1250 they would've spoken a Slavic language. However, 1250 was just after Mongol conquest of Kiev so the community could've been dispersed/disrupted (though that could make for a good question, a hypothetical Kievan Jewish refugee in Central Europe).
Of course later on Ukraine was Yiddish-speaking territory. That's because the Slavic- and/or Turkic-speaking Jewish communities were mostly dispersed by various wars (except in Crimea), and the Jewish population there dates from later, when Jews moved there from Poland. It's actually fairly late, since Jewish communities in Ukraine were largely dispersed/destroyed during the disastrous events of 1648, and the Jewish population of Ukraine was regenerated through migration/returning after that.
As a fun bonus, it's a little outside your time range, but there's evidence of communication between Jews in Kiev and Jews elsewhere, which is in Hebrew. It's a letter written in Hebrew to serve as an attestation that the bearer is in need of funds (a fundraising letter basically). While the letter itself is in Hebrew, the note at the bottom and some of the names suggest that the Jews there spoke a Turkic language.