I've changed wards several times in my life.
As I've gotten older and accumulated more practice, I've learned there's a few key points to making it a successful transition.
One, go to everything. Every Sunday meeting, every weekday activity, every service project. Seeing new people many times helps them feel less new (and scary,) and with the familiarity comes confidence and opportunities to strike up conversations.
Two, befriend the leadership, whether that's the bishop, Relief Society president, Sunday School teacher, or best yet, all of the above. These people are the people who know everyone, plus it's part of their job to be friendly. Their smiles and enthusiastic how-are-yous and introductions will help you maintain a minimum level of socialization while you find your niche in the ward.
Third, and related to the second, is to ask for a calling. Service is an excellent way to address loneliness, as it takes your focus away from your own needs and allows you to serve with others, thereby building the relationships and making the new ward feel like home.
My belief in these concepts solidified in the spring of 2011, and have served me well ever since. I had graduated from BYU-Idaho a few months previously, and I missed interacting with people my age with increasing desperation. Even though the women who had been my mentors growing up were just as friendly and helpful as ever, it just wasn't cutting it.
So, I started going to the singles ward. Within a few weeks I didn't feel much like "the new girl" anymore. I felt like I'd be missed if I didn't show, in part because I had responsibilities, but also because (faster than I could have hoped!) a circle of friends had formed around me. For being new, I felt awfully blessed.
This translated toward a magnanimous spirit as other new people came into the ward. Dani (the new new girl) and I quickly became officially-unofficial-best-friends, for the sake of having someone cheerful and sympathetic to sit by.
At the end of June, our sweet, perky Sunday School teacher invited us all to get into small groups to discuss a passage of scripture, with the idea that we were going to share the best insights from our group from the class. It's a very typical setup, and honestly, what happened next is pretty typical too.