Will picking missions randomly exclude me from doing other missions?

No it won't. There are 3 Missions that you won't be able to do as part of your storyline, BUT you can do them after you finish the mission you chose.

Picking missions randomly is okay, you just have to pick according to your style. Case in point: after zero hour, you get to choose between 'unlocking' firebats or marauders. depending on your playing style, you can choose to get firebats first.

Without spoiling too much, the 3 missions don't actually affect you that much. first one doesn't really matter because both unlock the same unit, second one give you a choice over 2 units, and the 3rd is a choice over which enemies you'll face.

All in all, if you just do missions randomly, eventually, you'll end up with only one mission left. That will advance you through the story line(I think if you choose that mission before doing the others, it will warn you about it)

Hope that helps. If you don't care about spoilers, here is an in depth guide for the campaign:

http://wiki.teamliquid.net/starcraft2/Campaign

In addition to all of these, there are some missions that will be easier/harder depending on which units/upgrades/mercs are available to you (e.g. Welcome to the jungle is easier if you have upgraded your missile turrets and stimpacks).

The campaign is basically a series of parallel storylines, that ultimately come together and lead into the final story. There is not really a set order to the campaign missions; any numbering on them is just one way to do them.

For example, it sounds like you are doing some of the missions with Tosh. His missions form a series. The "save the colonist" missions also form a series, along with the "find the artifacts" missions. You can progress on each story-arc independently, so you can basically do missions in whichever order you like.

The 3 missions that corroded mentioned in his answer are a different case. In 3 instances, you'll be given a choice of how to do something. Do you help person A, or person B? Each choice actually leads to a different mission. But as corroded also mentioned, you can still play the other mission, through the mission-replay function.

The only mission you need to do something special to unlock is the "secret" mission. You can learn more about that here: How do I get to the secret mission in Wings of Liberty? Otherwise, you can play the missions however you like and not miss anything. Which order you play them just determines the order you unlock units, and get money/research points.