Start dates are manipulated much in the same way due dates are using search operators to create SmartLists.

available search operators are:
[start/due]:
[start/due]Before:
[start/due]After:
[start/due]Within:

It's important to remember:

1) the best approach is to make the dates relative to today like the examples below. If you give an actual date, the SmartList date would have to be changed each time the search is run.

2) these operators don't look backwards. If you say:
[due/start]Within:"1 month of today"
the SmartList will only return tasks that
[start/are due] in the coming month, not tasks that
started or were due in the prior month..

Even the [start/due]Before:
search compares to the future:
[start/due]Before:today or [start/due]Before:"2 week of today".

Not possible: [start/due]Before/Within/After:"last week"

Here are some examples that could get you the tasks you seek via a Smart List.

startBefore:now AND dueWithin:"1week of today"
startBefore:today AND dueBefore:Saturday
startWithin:"1 month of today" OR dueBefore:"1 month of today"

A task with only a Start Date will not repeat daily as something can only start once. You can manually change the start date within the task or sweep them up via a smart list. To create the tasks you mention in your 3rd paragraph (start date but no due date) just use the start date search without a due date search and make the make sure your SmartLists pulls them in.

You can, however, create repeating tasks using the due date as reference.

On the web site when you create a task you have several icons you can click to quickly make the task (like Due, Priority, Tags, and so on). Unfortunately, there isn't a button to click for Repeat.

So, whether you are using the method or using SmartAdd, you click the "*" key and a repeating menu selection appears (every day, after week). Clicking any of these will continue repeat forever at the specified interval.

Once you click the * key, the last choice for repeating is "Advanced...". This allows you to say repeat XX number of times or repeat until a hard coded date.

Sorry this is long; hope it helps.

Start dates are optional and have no bearing on any of the default lists (Today / Tomorrow / This Week). As with all metadata (tags, locations, notes etc.) start dates help you group tasks for a particular purpose via the use of smart lists.

I use start dates like this; say I know that my passport is expiring on June 17th. I create a task called 'Passport expires today' and give it a due date of June 17th. Renewing a passport takes time, of course, so having it pop up on the 'Today' list on June 17th wouldn't be very useful. Instead, guessing that it'll take 6 weeks or so, I give it a start date of May 6th. I have a smart list called 'Upcoming' which is configured with the following filter:

((startWithin:"10 days of Today") OR (startBefore:today)) AND isRepeating:false

As you might imagine, it shows all tasks with a start date that occurs in the next ten days or in the past. I'm not interesting in seeing repeating tasks in the list so I filter those out but that's just personal preference. As part of my weekly review (http://bit.ly/1QqybIZ) I check that list. As early as April 26th the passport task will appear and, as soon as it does, I'll create a project to renew my passport and from that point it'll be on my radar. Once I've got my new passport I'll tweak the due and start dates of the original task appropriately and it'll disappear from the 'Upcoming' list until the next time.

Other task management applications use start dates in different (and potentially better) ways. I talk a little bit about that in the final post of this thread: http://bit.ly/1WUo03e

I'd like to respond to one of your comments and have quoted your original text below:

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So I set start date as Monday and due date as Friday I want it to appear on my to do list every day up to and including the due date. When I've done my work on it for that day I'll check it off and see a new task the next day.
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When you write 'to do list' are you talking about the 'Today' list? I'd suggest that all task management software works best if you create tasks that represent a single piece of work that you complete when it's actually finished. Having it pop up again the following day suggests that it isn't finished (and therefore probably shouldn't have been completed in the first place). Another quick example:

Let's pretend I'm going abroad shortly and have a number of things to do before I go away. This amounts to a project which I'll call 'Prepare for Trip to Spain'. I create a dedicated list to contain all the tasks, one of which is to call the credit card company to inform them of when and where I'm going so they don't block my card if I decide to use it. I create a task called 'Call credit card company' and tag it with '@call'. I'm going away on March 18th so I should probably have called them before then. To give myself a bit of leeway I set a due date of March 16th. I have a smart list that I use to group all the phone calls I need to make. This is the filter:

(tag:@call) AND NOT (dueAfter:"2 weeks of today" OR startAfter:today)

This list will show me all tasks tagged with '@call' whose start date is on or before today (which is to say, they've already started) and that are due within the next two weeks. I'm not worried about tasks due more than two weeks into the future, although of course they will pop up in due course. The list is sorted by due date so the tasks with due dates are at the top and those without sit at the bottom. On any given day I might decide I'm going to spend the next half hour making phone calls. Looking at the list I can see all the possible choices I have. If some of the deadlines are close then I might decide to make those calls first. Equally, if the earliest one isn't due for another week I might well decide my time is best spent making some of the other calls.

A massive post, I know, but I hope it helps. ;)

Cheers!

Thanks, azclaire and boikinton. Loads of great info there.

Just wanted to say that when you're creating a new task on the web app, there is an option for adding a repeat; it's the two arrows in a circle.