Stsadm.exe days are gone. Just for legacy purpose this commands are still compatible with SharePoint 2010, but let's see in this article how things have evolved after SharePoint 2007.
The first consideration that must be done before starting to use SharePoint's Powershell is that each command has it's own manual. So if you get stuck with a command, just type: man yourTrickyCommand. If you have just a little of Unix/Linux shell's commands background you should be already comfortable with this approach.

Let's start from the beginning by adding a solution into your farm

• SharePoint 2007: stsadm -o addsolution -filename "pathToSolutionFile.wsp"
• SharePoint 2010: Add-SPSolution "pathToSolutionFile.wsp"

and if you want to check the Add-SPSolution manual, just type  man Add-SPSolution, and you will receive the following output:

NAME
    Add-SPSolution

SYNOPSIS
    Uploads a SharePoint solution package to the farm.


SYNTAX
    Add-SPSolution [-LiteralPath]  [-AssignmentCollection ] [-Confirm []] [-Language ] [-WhatIf [
    ]] []

DESCRIPTION
    The Add-SPSolution cmdlet adds a SharePoint solution package to the farm. T
    his cmdlet does not deploy the uploaded SharePoint solution. Use the Instal
    l-SPSolution cmdlet to deploy the SharePoint solution in the farm.

    For permissions and the most current information about Windows PowerShell f
    or SharePoint Products, see the online documentation (http://go.microsoft.c
    om/fwlink/?LinkId=163185).

RELATED LINKS
    Get-SPSolution
    Update-SPSolution
    Uninstall-SPSolution
    Remove-SPSolution
    Install-SPSolution
    Remove-SPSolutionDeploymentLock

REMARKS
    To see the examples, type: "get-help Add-SPSolution -examples".
    For more information, type: "get-help Add-SPSolution -detailed".
    For technical information, type: "get-help Add-SPSolution -full".

Finally we have the full documentation into our shell!
There are two things that are really interesting:

• RELATED LINKS
• REMARKS

With the related links, we got all the informations needed to Install, Update, Delete, and with the remarks section we can see by examples how to definitely master the command. 
So if we need few examples, just type: get-help Add-SPSolution -examples

NAME
    Add-SPSolution

SYNOPSIS
    Uploads a SharePoint solution package to the farm.

    ------------------EXAMPLE------------------

    C:\PS>Add-SPSolution -LiteralPath c:\contoso_solution.wsp

    This example adds the SharePoint solution in the file contoso_solution.wsp
    to the farm.

Once you know how to obtain the PowerShell's commands documentation you are able to deepen by yourself.
I will just list you common PowerShell commands that are used mostly when deploying and installing SharePoint Solutions for a quick reference.

Solutions

Deploy a solution

• SharePoint 2007: sstsadm -o deploysolution -name SolutionFile.wsp -url http://yourServerUrl/ -immediate
• SharePoint 2010: Install-SPSolution -Identity SolutionFile.wsp -WebApplication http://yourServerUrl/

Upgrade a solution

• SharePoint 2007: stsadm -o upgradesolution -name SolutionFile.wsp -filename pathToSolutionFile.wsp
• SharePoint 2010: Update-SPSolution -Identity SolutionFile.wsp -LiteralPath pathToSolutionFile.wsp

Remove a solution

• SharePoint 2007:  stsadm -o deletesolution -name SolutionFile.wsp
• SharePoint 2010: Remove-SPSolution -Identity SolutionFile.wsp

And finally let's see the all process from installing to uninstalling a SharePoint Solution by examples:

• Add-SPSolution c:\path\to\SharePointSolution.wsp
• Install-SPSolution -Identity SharePointSolution.wsp -WebApplication http://sharepoint2010 -GACDeployment
• Uninstall-SPSolution -Identity SharePointSolution.wsp -WebApplication http://sharepoint2010
• Remove-SPSolution -Identity SharePointSolution.wsp