Osaka Station to KIX

I need to be at KIX (Terminal 2) at 09:00am. What is the best connection from Osaka Station to KIX? Bus or Train?

It is personal preference. Bus or train are possible.

The advantage of the bus is that you are guaranteed a seat and that some of them go all the way to the terminal 2 building. The disadvantage is that they are more prone to get stuck in traffic jams.

The advantage of the train is that it is cheaper and more reliable, but you may have to stand. Also, it is quite a distance from the airport station to the terminal 2 building.

I need to be at KIX (Terminal 2)

So, you are going to use a low cost carrier [LCC] from KIX, right?

Airport Limousine buses may be available, but if you travel on a weekday / in a busy season, I suppose a railroad route can be better, considering a possible congestion on road.

In KIX, there are shuttle buses connecting Aeroplaza (1F) and Terminal 2 within 10 minutes.
Leaving Osaka Station around 07:00 sounds enough to reach KIX Terminal 2 by 09:00 by rail and shuttle bus.
http://www.kansai-airport.or.jp/t2/en/access/access05.html
http://www.kansai-airport.or.jp/t2/access/images/bus_info.pdf

From Osaka Station you can take one of the Airport Rapid trains to Kansai-airport Station (for about 70 minutes), paying 1190 yen per adult.
To be precise, these Rapid trains are two-destination trains bound for Kansai-airport and Wakayama.

If you hope to take a higher-grade train on most of your way, you can
first take a Local train on Osaka Loop Line's outer track to Tennoji Station (for about 20 minutes)
then catch a Ltd. Express "Haruka" train to get to Kansai-airport Station (in about 35 to 40 minutes),
paying 1840 yen per adult if you do not use a "Haruka" reserved seat or 2160 to 2560 yen per adult if you use one.
You can try reserving your "Haruka" seat at a JR ticket counter.

Alternatively you can take a Local train on Osaka Loop Line's inner track, make a transfer in Shin-Imamiya to Nankai and catch a Ltd. Express "Rapi:t Beta" train. You will have a staircase to go up during the transfer, so this route sounds good for a backpacker. The one-way cost will be 1610 yen per adult.
Unlike "Haruka" trains of JR-West, "Rapi:t Beta" trains of Nankai have seats only to be reserved.
You can try reserving your " Rapi:t Beta" seat at a ticket counter in Namba Station of Nankai.

Thank you all for your information. Yes, I am taking a low-cost carrier.

My main concern is to have less hassle in finding the train connections.

I would like to consider taking a higher-grade train on Osaka Loop Line's outer track to Tennoji Station (for about 20 minutes)
then catch a Ltd. Express "Haruka" train to get to Kansai-airport Station (in about 35 to 40 minutes). Does the Express Haruka train stops at Terminal 2?

Thank you.

Let me clarify:
Local trains on Osaka Loop Line are normal urban trains used by passengers such as commuters.
I used the word "higher-grade�h for Ltd. Express trains such as "Haruka�h and "Rapi:t Beta."

Unfortunately there are no trains running to/from KIX Terminal 2.
If you travel on a weekday it may be better to leave Osaka Station earlier in order to avoid crowdedness in the station and in a Local train.
In Tennoji Station of JR-West, probably your Local train (on Osaka Loop Line's outer track) will arrive to Track 14, 17 or 18 and your "Haruka" train will depart from Track 15. Each of the platforms for these four tracks is equipped with escalators and an elevator. You can change platforms via a passenger bridge.
http://www.jr-odekake.net/eki/img/premises/0620831.pdf

Thank you very much for your prompt and detailed reply. It is indeed very helpful.