Hello Alexbarsa1
I normally lurk on this site and don't post. Your post reminds me of where I was a while back. I assume you already have a trading plan which includes the set-ups you want to use on which particular criteria of stocks etc etc. You have studied patterns, times and market tick etc. Notice also you want to play NYSE as well as NASDAQ stocks (be careful with NYSE specialists ;-) they burn the inexperienced and experienced alike!
Although I belong to a USA cyber trading room, I also select some stocks of my own that look like they may be beginning to get institutional interest and thereby could give decent potential. I use simple set-ups including breakouts. My own daily watchlist has around 15 stocks which can change daily if there isn't much follow-through (like recently). Typically I will trade stuff GOOG and SHLD as they make my profit most days. It works ok for me, nothing too complicated really.
I find stocks using the free screener in YAHOO finance. Run a scan at the end of each day using your own criteria and up pops the names. Analyse the charts for these, check news, upgrades etc. for reasons for the moves and put them into categories. Won't say more about that as I don't know what you are after. Personally I stick with the categorization described by Weinstein in his book (for investors really but we can go after the same stocks). You could equally use the categorization described by O'Neil. In essence I'm after stocks breaking out of a long-term base which could go to the moon but the money I'm taking daily is only pin money from the MM's and institutions point of view.
Eventually you will find which stocks are working best for you. Don't go after stuff that is too thin or stuff that daytraders are killing themselves over, at least initially until you get used to the whole thing.
The YAHOO screener has a realtime version, I think for around 10 bucks per month. Realistically though, to begin with, start the day with a small list and at least a couple of them should give you what you want. Don't know if you really need to be running scans in the beginning, chasing after stuff which may have already moved can be dangerous until you get used to the markets and the way US stocks move. You can scan for stuff which is reaching a particular point such as 52 week high/low etc.
Best advice I can give you however from my own experience and disappointments, buy and read a few good books about trading and investing, it'll give you the basic understanding of the market you will need. If you have already done that I personally went the education route given in the USA trading room I use to be the best value for money available (year subscription for less than the cost of some one day courses in the UK!!!). I have talked to other traders using other rooms who have also found them to be great value. I don't think I can name them here however surf the net and you will find what you are looking for. One word of caution, avoid the rooms that are calling the trades and use a room that is teaching you how to trade instead.