My cleric wants to learn  how to use a herbalism kit. How long would you make the time it takes to learn it, along with how much gold is required. Any checks to speed things up? 

When I have someone who wants to learn a tool proficiency I can handle it in a couple of different ways.  It kind of depends on if the party is actively moving around or if they're in an in-game break, or perhaps working out of a city and doing radial quests.

If they are actively adventuring in non-civilized areas, it is some experimentation with live ingredients.  You could roll with either Int or Wis, kind of your call.  If it's a Cleric, and they're wanting to learn about medicinal herbs, go Medicine check so that it's based on their Wisdom (if you want to be nice).  Nature makes equally as much sense, but then your Cleric may have dumped Intelligence a little.  Again, all your call on that.  As to the cost, I would say (obviously) the cost of the kit, plus some additional cost for things like sample vials and the sort prior to them heading out into the wilderness.

If the party is in a town or city, whether for a break or as an adventure hub/base of operations, they could spend some of their downtime with a local artisan/merchant that knows something about Herbalism already.  The cost of this would be mostly the time spent with the NPC, but you may also need to include additional costs for ingredients that the NPC will be out from the teaching process.  Checks would be similar to above, kind of your call.

Finally, if the PC wants to purchase a book on Herbs and Herbalism, that would help.  It could bring down the DC of checks, or reduce the number of checks required.  This would, also, add some gold cost to the overall process.

Something else to consider is if the place they're at even has kits; some places that sell books may be out of the kits.  I have a character that recently had to settle for buying what amounts to a "How-To" book on building various tool kits.  So while I am able to gain proficiency in Alchemy, I have to find the parts to make a kit unless we come to a town which sells them whole.  It's just another way to go about pacing the advancement of a character.  It allows them to move forward at a pace, but not just immediately learn and get everything all at once.  Plus, it may help make a player who normally just coasts through become a little more involved as they will be actively seeking things.

I would suggest looking at XGtE's rules for training during downtime. These are fairly straight forward rules you can expand upon or keep very minimal.

Training

You can spend time between adventures learning a new language or training with a set of tools. Your DM might allow additional training options.

First, you must find an instructor willing to teach you. The DM determines how long it takes, and whether one or more ability checks are required.

The training lasts for 250 days and costs 1 gp per day. After you spend the requisite amount of time and money, you learn the new language or gain proficiency with the new tool.

Basic Rules >> Adventuring >> Between Adventures >> Downtime Activies >> Training