How to Align Lightning Component Label and Fields

Yes, simply place your elements in a horizontal form, and give the elements the correct size. Here's a modified example of your code:

<div class="slds-form_horizontal slds-form slds-grid slds-wrap">
    <lightning:input aura:id="contactField" class="slds-size_1-of-2" type="text" value="{!v.newContact.LastName}" Name="Last Name" label ="Last Name"/>
    <lightning:input aura:id="contactField" class="slds-size_1-of-2" type="text" value="{!v.newContact.FirstName}" Name="First Name" label ="First Name"/>
    <lightning:input aura:id="contactField" class="slds-size_1-of-2" type="email" name="email" label="Email"  value="{!v.newContact.Email}" />
</div>