It doesn't help you much to have thousands of followers if they never engage with your tweets. You need to make them interested, and the easiest way to do that, is to make them notice you. Tweeting often can help you get noticed, and help you become a familiar face to your followers.
Spamming can have the opposite effect on your followers, so you need to somehow make up quality tweets. Something of a high enough quality to get noticed – retweets and quotes is generally of poor quality. Scheduling your tweets can save you some time, the disadvantage is that you likely don't see if anyone responds to your tweets.
How often should you tweet?
The more time you spend tweeting the better, but you will be limited by the material that you got to tweet – if you don't have anything to tweet, you shouldn't resort to re-tweets and posting quotes like some spam accounts are doing. The followers you gain trough spamming are more likely to be automated bot accounts than real followers, this is because they are following people by keyword.
How often you should tweet is also limited by twitter – to avoid getting banned, you should stay within the daily limits. It likely won't do any harm to occasionally hit a limit, but you shouldn't be doing it every day. A Quote from the Twitter help center says:
Tweets: 1,000 per day. The daily update limit is further broken down into smaller limits for semi-hourly intervals. Retweets are counted as Tweets.
See also (external): About Twitter Limits (Update, API, DM, and Following)
Reach the full potential of your twitter account
Some people recommend not to follow more than a couple of hundreds users per day, or only send out a few tweets per day, but there is nothing indicating that you get banned for following hundreds accounts – abusing twitter for spam and other malicious behavior can get you banned!
Pushing the daily limits of twitter should not get you banned, unless you are found to be doing it with malicious intentions. We'd say, staying within daily limits, without hitting them every day, shouldn't get you into to much trouble.
Posting quality tweets
The quality of your tweets is very important when working with twitter, having thousands of followers is of little value if they just ignore your tweets. There are many fake accounts on twitter, so many that people have gotten used to them – your task will be to appear as real and original as possible!
Your main goal will be to get your followers to pay more attention to your tweets, so if you can't meet a certain level of quality, then its likely better to wait with the tweeting until you have something to tweet about again. Its also fine to chat with people, but even your chats should be of a certain quality to appear interesting to your other followers.
If you decide to sell tweets on twitter, you should also be careful about what you accept to tweet.
See also: Selling tweets on twitter