Run Death House
Death House serves much better as an introduction to CoS over Phandelver. Phandelver is a great adventure, and I started as a player with it, but it would serve you better later if you want to run a classic D&D adventure, with less gothic horror elements. I'm DMing Curse of Strahd now and started my group with Death House.
Death House will teach you certain DM skills, and it will reinforce the skills necessary for the PCs to survive Curse of Strahd. Here's my justification:
Introduce with Roleplay and dread
Curse of Strahd is a roleplay-heavy campaign — there are lots of situations you don't want to bash your way through. Additionally, Death House will, if played correctly, teach you as DM to foreshadow threats and create a sense of dread that will help with the rest of the campaign on both sides of the table.
For example, the desolate streets of Barovia, with only a single house on the outskirts lit up, with smoke pouring out of the chimneys will draw the players in, but also give them the eerie feeling something is wrong. You'll have to play the children correctly to bait the players in, but you can always use the mists to railroad them as necessary. Once they're in the house, you'll be able to hook them in.
This introduction is much more in-line with the tone and nature of the rest of the campaign.
Take time to describe the environment
This is critical with Death House, but less emphasized with Phandelver. With Death House, you'll need to take time in each room, describing the blazing hearths, the strange wolf-helm armor, everything. It's all important, because it also teaches the players to pay attention to the environment. When
the blazing hearths pour out poison smoke in the 2nd phase,
they'll get a good idea of how Barovia works. You'll have plenty of chances to have them roll Investigation, Perception, and Insight during the buildup, and therefore teach them some fundamental rules, and fundamental survival skills for later.
Toss a bit of combat in after
The first fight will likely occur later,
on the 3rd floor.
This will be after the players have had the run of the house for a bit, and will give you plenty of time to switch from warm, but empty house, to
the dusty, old, decrepit 3rd floor.
Using that buildup, when something bad happens like
the armored statue punching a player in the face,
the players will be alert to such changes. It's also an easy fight to run, but will introduce damage resistance and basic combat rules.
Phandelver starts you off with a goblin ambush, which is fine, but Curse of Strahd places the emphasis on the fact that the players can walk into dangerous areas — Phandelver's goblin ambush is more of a device used to spur the plot along (perhaps much like the mists). Death House will give you a chance to explore first. Having the combat after emphasizes where the priorities should be in this campaign, I think.
Ramp up the danger
Death House is notorious for how lethal it can get — your players can absolutely TPK against
the Shadows, or the Shambling Mound in the basement.
However, much of the campaign is like this — you'll get to make the decision of how hard to beat them over the head with their impending doom, or how to fail forward. You'll learn from your players what they expect out of a campaign with this kind of tone, and it'll get everyone on the same basis. Starting with Phandelver, I think you stand the danger of letting them be the big damn heroes and then taking that away from them.