Besides the painting suggested by Chad of Lonestar Painting which can truly makeover the appearance of a home, I suggest that the trees and shrubs be professionally trimmed and thinned out.  Application of weed control and fertilization of the lawn will provide a healthy and attractive canvas to your yard that can be supplemented with a dose of spring color annuals

A very quick and cost effective way to upgrade your homes exterior is a cleaning and color change. Pressure wash to clean it up and paint the shutters and front door.  They can remain the same color or pick a fresh new color.  You can also take any old fixtures that have faded over time and paint the metal with a fresh satin or gloss paint.  Same with metal railings or metal awnings. Possibly a one day job with big results. 

We just had sod installed at the office here at Crown Builders in Charlotte, NC.  The landscaper told me to water each day for 15 minutes the first 10 days.  From 10-21 days after install water every other day.  After 3 weeks and beyond cut the lawn as needed.  It should be fairly well rooted and can take normal wear and tear. Give it 1" of water each week.  Apply balanced turf fertilizer 6 weeks after install.

Depending on the severity of the runoff you might want to contact a Civil Engineer. As an engineer myself, with a specialty in drainage and hydrologythere are many issue to consider when dealing with offsite drainage. For instance if you redirect it on to another property it's possible you could be held liable for any impacts tha occur. Just a thought.

We agree with Geoff's suggestion and would also recommend speaking with a local landscape architect. They not only handle plant specifications, but also deal with drainage and irrigation. They will be able to assist you or point you in the right direction to solve your issue.  

A landcaping contractor might help. Depending on your location, you may be able to find someone here: http://www.guildquality.com/find/Landscaping-and-hardscaping/North-America

You can install a flexible downspout extension on your downspout that is next to your front step and run it behind your bush out past your mulch to keep the water from washing your mulch away.  You can buy the flexible downspout extensions at Lowes or Home Depot.

What is the water source? It looks like it is coming from a drain pipe behind a shrub? I would recommend relocating that to dump to a different location. No matter what type of edging you install on the bed with the mulch, a massive amount of water from that down pipe will cause some overflow and mess. Maybe you can incorporate a stone trail where the water flows, if you can't relocate the down pipe. Ideally, I'd want to see the drain pipe go under the side walk, and out by the grass, or if its by a garage, down the driveway.

Emerald Green Arborvitae

Install a decorative stone border or edging that will help hold the mulch in place.