Hey Buddy, Yeah doing slayer at the same time is ALWAYS a good idea You should obviously start off with all no-memb quests and then try to remember any other quests you had done on your old main account and do them aswell^^
It's nice to see you show some interest back into the RuneScape game. As far as advice and your question concerning slayer and combat. Of course you should train both and the same time, doing this benefits your account so much; and here's how:
1. For starters, you will gain combat experience in the various skills of Attack, Strength, Defence, Magic, Ranged and of course Hitpoints. 2. You will see an increase in your supply of various items, meaning you'll get items such as herbs, runes, etc. 3. Not only will your combat experience go up along with your combat level but you will also see your Slayer level go up which is good because it's nice to have a Slayer helmet earlier than expected; which costs 400 slayer points and needs 55 crafting to assemble it and it can also be imbued at Nightmare Zone.
I would recommend you do some quests first to give you some quick xp and access to various slayer monsters/areas. The waterfall quest, fight arena and Tree Gnome Village are some great starters. Once you have a stable base of combat level ~50, then I would start training slayer to make it quicker and more efficient.
Welcome Adventurer! Feel free to Add me there is way too much to type out here and so many directions you can go! that is the lovely aspect of runescape i suppose. If you arent interested in contacting me, then my advice is quests all the way up to dragon slayer / heros at bare minimum. May Zamorak Be with you, -FreeAgent