It is extremely easy to start the romance subplot in ME1. Normally you just have to talk to the person you're interested in after each story mission.
The mass effect wiki has a nice breakdown of the specific conversations: http://masseffect.wikia.com/wiki/Romance
Also note that unless you want to get a confrontation where you will have to choose either Liara or Ashley you should refrain from talking to the character you do not want to romance about anything personal. --- Straight to the pointless.