Russia knows that Trump is a strong leader. He's not going to back down or apologize or have more flexibility after the election, like Obama did. Russia has already stated, since the election, that he wants to mend ties with the United States. It's important that they develop a plan to fix what's wrong in Syria.
While Trump's attention on Russia and Putin since his election win has been highly controversial, I do support the relationship he is trying to foster with Russia. Russia and the United States have an incredibly rocky relationship, and it is time for he political wounds to be mended. Russia is a huge superpower, and having them as an ally would certainly strengthen the United States in many ways.
Trump has talked with Putin a lot, and has also talked with many other world leaders. The media is trying to turn Trump's discussions with Putin into something significant, but it is not particularly worrisome. Trump is a smart and savvy business person and will ultimately form alliances that are in the best interest of America.
Many world leaders communicate with Putin on a daily basis. Examples of such are German Chancellor Angela Merkel and British Prime Minister Theresa May; this is due to the role these two nations play within the European Union and the North Atlantic Treaty Organisation; respectively.
But in regard to the question posed, no. Trump's relationship toward Putin is one of appeasement above all else. Putin has a reputation for being a warmonger (just look to the Ukraine or the conflict in the Caucasus), and furthermore, he had already made his intentions clear that he desires to restore the borders of the former USSR.
Trump has stated, time and time again, that he respects Putin due to him being a 'strong leader'; this is in the sense that Putin has proved himself to be a violent dictator who routinely instigates purges on the scale of Stalin.
So no, appeasing this type of individual is not the way to go. It didn't work in trying to get Hitler to cease his warmongering ways, and it will not work in ceasing Putin.
No, I do not support Trump's relationship with Russia. Having good cordial relationships with foreign leaders is one thing. Openly praising a dictator who often acts in opposition to the United States is another. The fact that Russia seems to be so tightly connected to hacking and interference with the election also leads to some "huge" question marks about Trump's relationship with Putin.